CHAPTER 14

Number Properties on the GMAT

Each of the 600 elements of Set X is a distinct integer. How many of the integers in Set X are positive odd integers?

(1)   Set X contains 150 even integers.

(2)   70% of the odd integers in Set X are positive.

Above is a typical Data Sufficiency question dealing with number properties. In this chapter, we’ll look at how to apply the Kaplan Method to this question, discuss the number properties rules being tested, and go over the basic principles and strategies that you want to keep in mind on every Quantitative question involving number properties. But before you move on, take a minute to think about what you see in this question and answer some questions about how you think it works:

PREVIEWING NUMBER PROPERTIES ON THE GMAT

Number properties are all about categories and rules. Certain kinds of numbers behave the same ways in all cases. When discussing number properties, don’t assume that they apply exclusively to integers. Integers are one subset of numbers: namely, positive and negative whole numbers and zero. The concept of number properties is perfect fodder for the GMAT testmakers because it allows them to reward you for using the Core Competencies of Pattern Recognition and Critical Thinking to draw inferences about how numbers behave, based on certain characteristics they possess. That’s why number properties questions appear on the Quant section with greater frequency than other topics: these questions constitute approximately 20 percent of GMAT Problem Solving questions and nearly 30 percent of Data Sufficiency questions.

What Does It Mean That Each of the Elements Is a Distinct Integer?

Distinct simply means “different.” So in this question, all 600 integers are different—there are no integers that show up more than once in the set. Don’t assume that a set of integers always contains only distinct integers. This concept often comes into play with questions dealing with variables. Just because x and y are different letters does not necessarily mean that they represent different values.

How Are Integers Related to Even and Odd Numbers?

All integers are either odd or even. Students often forget that 0 is an even number. All integers ending with the digit 0, 2, 4, 6, or 8 are considered “even numbers,” and all integers ending with the digit 1, 3, 5, 7, or 9 are considered “odd numbers.” When a number is classified as even or odd, it must also be an integer; non-integers are not considered odd or even.

How Are Positive Integers Different from Non-negative Integers?

Basically, the only difference is the number zero. Zero is the only number that is neither positive nor negative. This is a great example of how number properties questions test your Attention to the Right Detail. A question that asks about negative numbers does not include zero as a possibility, but a question that asks for non-positive numbers does include zero. You can bet that considering zero when it shouldn’t be considered, or vice versa, can lead to a wrong answer on Test Day.

What GMAT Core Competencies Are Most Essential to Success on This Question?

As mentioned earlier, because number properties deal with the expected behavior of numbers based on certain rules, Pattern Recognition is an essential Core Competency for these questions. Also, you should use Critical Thinking in two ways: by remembering the abstract rules of number properties and by using the Kaplan strategy of Picking Numbers to make those properties more concrete in your mind on Test Day.

Here are the main topics we’ll cover in this chapter:

Handling GMAT Number Properties

Now let’s apply the Kaplan Method to the number properties question you saw earlier:

Each of the 600 elements of Set X is a distinct integer. How many of the integers in Set X are positive odd integers?

(1)   Set X contains 150 even integers.

(2)   70% of the odd integers in Set X are positive.

Step 1: Analyze the Question Stem

This is a Value question. To have sufficiency, you need a single value for the number of positive odd integers in Set X. The question stem tells you that the total number of integers in the set is 600 and that they are all distinct, or different from each other.

Step 2: Evaluate the Statements Using 12TEN

Once you are sure you understand the information in the question stem, move on to the statements. Statement (1) tells you the number of even integers in Set X, from which you could determine the number of odd integers in Set X. But you don’t know the number of these odd integers that are positive. Statement (1) is insufficient. Eliminate (A) and (D).

Statement (2) tells you the percentage of the odd integers in Set X that are positive. But Statement (2) tells you nothing about the number of odd or even integers in Set X. Statement (2) is insufficient. Eliminate (B).

Now combine the statements. From the two statements together, you could determine the number of odd integers and the percentage of those odd integers that are positive. Together, the two statements are sufficient. Choice (C) is correct.

Now let’s look at each of the six areas of number properties that show up on the GMAT Quantitative section, starting with integers and non-integers.

INTEGERS AND NON-INTEGERS

Integers are a particularly useful number properties category for the GMAT testmakers, since questions that focus on the rules governing integers force test takers to discriminate between different categories of numbers (whole numbers versus fractions or decimals) and since integers include positives, negatives, and zero. Additionally, integers can be easily combined with other number properties to make more difficult questions (e.g., saying that the square root of a number is an integer means that the number is a perfect square, or saying that the quotient of two numbers is an integer means that the numbers are multiples/factors of one another).

These questions also contain an important trap that you must learn to avoid: never assume a number is an integer unless you’re told that it is. The absence of information in a GMAT question can be just as important as its inclusion.

Real Numbers

All numbers on the number line. All of the numbers on the GMAT are real.

Integers

All of the numbers with no fractional or decimal parts: in other words, all multiples of 1. Negative numbers and 0 are also integers.

Rational Numbers

All of the numbers that can be expressed as the ratio of two integers (all integers and fractions).

Irrational Numbers

All real numbers that are not rational, both positive and negative (e.g., π, −image).

On the GMAT, it’s highly unlikely that you’ll get a question that uses the terms rational or irrational, but you will see many questions that use the term integer. Both positive and negative whole numbers are integers. Zero is also an integer. Keep in mind that if a question doesn’t say a number is an integer, then the number could be a fraction. Some Data Sufficiency answers depend upon this possibility.

Two rules are important to remember when performing operations with integers:

As with all number properties questions, Picking Numbers for questions about integers and non-integers can make them easier to tackle.

In-Format Question: Integers and Non-integers on the GMAT

Now let’s use the Kaplan Method on a Data Sufficiency question dealing with integers and non-integers:

Is z an integer?

(1)   2z is an even integer.

(2)   4z is an even integer.

Step 1: Analyze the Question Stem

This is a Yes/No question, so remember that either “always yes” or “always no” is required for sufficiency. The stem asks you whether z is an integer. It doesn’t provide any other information, so move on to the statements.

Step 2: Evaluate the Statements Using 12TEN

Statement (1): If 2z is an even number, z must be an integer because all even numbers can be evenly divided by 2. You can use Picking Numbers to test this. For instance, if 2z = 2, then z = 1. If 2z = −122, then z = −61. You can pick any even integer for 2z and always find that z is an integer, so Statement (1) is sufficient. Eliminate (B), (C), and (E).

Statement (2) looks similar to Statement (1), but you can use Picking Numbers to be sure. If 4z = 4, then z = 1, which is an integer. But if 4z = 6, then z = 1.5, which is not an integer. So you can’t say that z is always or never an integer. Statement (2) is insufficient, and (A) is correct.

TAKEAWAYS: INTEGERS AND NON-INTEGERS

 

Practice Set: Integers and Non-integers on the GMAT

Answers and explanations at end of chapter

  1. Is z an integer?

    1. (1)   image is an integer.

    2. (2)   image is NOT an integer.

  2. If x and y are positive integers, is image an integer?

    1. (1)   x + y = 30

    2. (2)   image

  3. If d is a positive integer, is image an integer?

    1. (1)   image is an integer.

    2. (2)   image is NOT an integer.

ODDS AND EVENS

The terms odd and even apply only to integers. Even numbers are integers that are divisible by 2, and odd numbers are integers that are not. Odd and even numbers may be negative; 0 is even. A number needs just a single factor of 2 to be even, so the product of an even number and any integer will always be even.

The GMAT tests the same odds and evens rules over and over. The rules are simple, and putting a little effort into memorizing them now will save you precious time on Test Day.

Rules for Odds and Evens

Odd ± Odd = Even

Odd × Odd = Odd

Even ± Even = Even

Even × Even = Even

Odd ± Even = Odd

Odd × Even = Even

Applying these rules, we notice some important implications for odd and even numbers raised to exponents:

Exponent Rules for Odds and Evens

Odd any positive integer = Odd

Even any positive integer = Even

Knowing these rules cold will free up your mind so you can strategize when answering questions. As with all number properties questions, Picking Numbers is a useful strategy to use for questions that deal with odds and evens.

In-Format Question: Odds and Evens on the GMAT

Now let’s use the Kaplan Method on a Data Sufficiency question dealing with odds and evens:

If x is an integer, is x odd?

(1)   x + 4 is an odd integer.

(2)   image is NOT an even integer.

Step 1: Analyze the Question Stem

This is a Yes/No question. You are told that x is an integer. You want to determine whether there is enough information to answer the question “Is x odd?” Now look at the statements.

Step 2: Evaluate the Statements Using 12TEN

Statement (1) says that x + 4 is an odd integer. So x is equal to an odd integer minus 4. Because 4 is an even integer, x is an odd integer minus an even integer. Because an odd minus an even is odd, x must be odd. Statement (1) is sufficient. You can eliminate (B), (C), and (E).

Statement (2) says that image is not an even integer. To determine sufficiency, pick a number for x.

If x = 3, then image is an odd integer, so image is not an even integer, and Statement (2) is true. In this case, x = 3 is odd, so the answer to the question is “yes.”

If x = 2, then image is not an even integer; in fact, it is not an integer at all. And that is fine, because nowhere in the question does it state that image is an integer. So Statement (2) is true. In this case, x = 2 is even, not odd, and the answer to the question is “no.”

Because more than one answer to the question is possible, Statement (2) is insufficient. Choice (A) is correct.

TAKEAWAYS: ODDS AND EVENS

 

Odd ± Odd = Even

Odd × Odd = Odd

Even ± Even = Even

Even × Even = Even

Odd ± Even = Odd

Odd × Even = Even

Practice Set: Odds and Evens on the GMAT

Answers and explanations at end of chapter

  1. If z is an integer, is z even?

    1. (1)   image, where m is an integer.

    2. (2)   z3 is even.

  2. The set S contains n integers. Is the sum of all the elements of set S odd?

    1. (1)   All the elements of S are prime numbers.

    2. (2)   n = 2

  3. If negative integers k and p are NOT both even, which of the following must be odd?

    • kp

    • 4(k + p)

    • k − p

    • k + 1 − p

    • 2(k + p) − 1

POSITIVES AND NEGATIVES

Some GMAT questions hinge on whether the numbers involved are positive or negative. These properties are especially important to keep in mind when Picking Numbers on a Data Sufficiency question. If both positives and negatives are permissible for a given question, make sure you test both possibilities, since doing so will often yield noteworthy results. Take the same approach that you’ve been learning to use for other number properties: spend some time memorizing the rules but always keep your eye out for strategic opportunities.

Properties of Zero

Adding or subtracting zero from a number does not change the number.

Examples:   2 + 0 = 2

 4 − 0 = 4

Any number multiplied by zero equals zero.

Example:     12 × 0 = 0

Division by zero is undefined. When given an algebraic expression, be sure that the denominator is not zero. The fraction image is likewise undefined.

Properties of 1 and −1

Multiplying or dividing a number by 1 does not change the number.

Examples:   4 × 1 = 4

−3 ÷ 1 = −3

Multiplying or dividing a number by −1 changes the sign, but not the absolute value.

Examples:   6 × (−1) = −6

−2 ÷ (−1) = −(−2) = 2

Note: The sum of a number and −1 times that number is equal to zero.

The reciprocal of a number is 1 divided by the number. The product of a number and its reciprocal is 1. Zero has no reciprocal, since image is undefined.

Properties of Numbers between −1 and 1

The reciprocal of a number between 0 and 1 is greater than the number itself.

Example:  The reciprocal of image is image.

image

(You can also easily get the reciprocal of a fraction by switching the numerator and denominator. The result will be the same. For example, the reciprocal of image is image.)

The reciprocal of a number between −1 and 0 is less than the number itself.

Example:  The reciprocal of −image is image

image

The square of a number between 0 and 1 is less than the number itself.

Example:  image, which is less than image.

Multiplying any positive number by a fraction between 0 and 1 gives a product smaller than the original number.

Example:  image, which is less than 6.

Multiplying any negative number by a fraction between 0 and 1 gives a product greater than the original number.

Example:  image, which is greater than −3.

The special properties of −1, 0, and 1 make them important numbers to consider when you are Picking Numbers for Data Sufficiency questions, as well as for the “could be/must be” kinds of Problem Solving questions. Because numbers between −1 and 1 can make things larger or smaller in different ways than do other numbers, they are good numbers to pick when testing whether one expression always has to be less than or greater than another.

All these properties can best be seen by observation rather than by memorization.

Operations with Positives and Negatives

Addition

With like signs: Add the absolute values and keep the same sign. Adding a negative number is the same as subtracting a positive number.

Example:  (−6) + (−3) = −9

With unlike signs: Take the difference of the absolute values and keep the sign of the number with the larger absolute value.

Example:  (−7) + (+3) = −4

Subtraction

Subtraction is the inverse operation of addition, so subtracting a negative number is the same as adding a positive number.

Example:  (−5) − (−10)

= (−5) + (+10)

= 5

Multiplication and Division

The product or the quotient of two numbers with the same sign is positive.

Examples:  (−2) × (−5) = +10 and image

The product or the quotient of two numbers with opposite signs is negative.

Examples:  (−2) × (+3) = −6 and image

Keep in mind that it doesn’t matter whether the negative sign is in the numerator or in the denominator. The fraction image is the same as image and image.

In-Format Question: Positives and Negatives on the GMAT

Now let’s use the Kaplan Method on a Data Sufficiency question dealing with positives and negatives:

Is x − 2y + z greater than x + y − z?

(1)   y is positive.

(2)   z is negative.

Step 1: Analyze the Question Stem

This is a Yes/No question. The question stem asks, “Is x − 2y + z > x + y − z?” Don’t miss an opportunity to use Critical Thinking to find a more efficient solution to this problem. You can simplify this inequality to make it easier to work with:

x − 2y + z > x + y − z

−2y + z > y − z

z > 3y − z

2z > 3y

This simplified inequality is equivalent to the original inequality. Now you can work with the equivalent question stem, “Is 2z > 3y?” Look at the statements next.

Step 2: Evaluate the Statements Using 12TEN

Statement (1) says that y is positive. However, you’re given no information about z, so this statement is insufficient. You can test this by Picking Numbers. For example, if y = 1 and z = 5, then y is positive, so Statement (1) is true. You have 2z = 2(5) = 10 while 3y = 3(1) = 3. In this case, 2z is greater than 3y, and the answer to the question is “yes.”

However, if y = 1 and z = −4, then y is positive, so Statement (1) is true. You have 2z = 2(−4) = −8 while 3y = 3(1) = 3. In this case, 2z is less than 3y, and the answer to the question is “no.”

Because more than one answer to the question is possible, Statement (1) is insufficient. You can eliminate (A) and (D).

Statement (2) says that z is negative. However, you have no information about y, so this statement is insufficient. For example, if you pick numbers so that y = −4 and z = −1, then z is negative, so Statement (2) is true. You have 2z = 2(−1) = −2, while 3y = 3(−4) = −12. In this case, 2z is greater than 3y, and the answer to the question is “yes.”

However, if y = 1 and z = −4, which are the values that you worked with when you considered Statement (1), then z is negative, so Statement (2) is true. Again, 2z = 2(−4) = −8 while 3y = 3(1) = 3, so in this case, 2z is less than 3y, and the answer to the question is “no.” Because more than one answer to the question is possible, Statement (2) is insufficient. You can eliminate (B).

Taking the statements together, as y is positive, 3y is positive. Because z is negative, 2z is negative. Because 2z is negative and 3y is positive, 2z is less than 3y. It is not true that 2z > 3y. You can answer the question with a definite “no.” The two statements taken together are sufficient. Answer choice (C) is correct.

TAKEAWAYS: POSITIVES AND NEGATIVES

 

Practice Set: Positives and Negatives on the GMAT

Answers and explanations at end of chapter

  1. If x − y = 8, which of the following must be true?

    1. Both x and y are positive.

    2. If x is positive, y must be positive.

    3. If x is negative, y must be negative.

    • I only

    • II only

    • III only

    • I and II

    • II and III

  2. If x and y are integers such that x > y, which of the following CANNOT be a positive integer?

    1. y(y − x)

    2. y(x − y)

    3. image

    • I only

    • II only

    • III only

    • I and II only

    • II and III only

  3. Is x > y?

    1. (1)   9x = 4y

    2. (2)   x > −y

FACTORS AND MULTIPLES

Factors and multiples are relatively simple concepts used to make some fairly difficult GMAT problems. Multiples—the products of a given integer and other integers—boil down to the numbers you list when “counting by” a certain number. For example, the multiples of 5 are 5, 10, 15, 20, and so on. Factors—those integers that divide another integer without leaving a remainder—are a little trickier, but this section and the next, on remainders and primes, will show you some ways to determine the factors of a number on Test Day.

Multiples

A multiple is the product of a given integer and another integer. An integer that is divisible by another integer without a remainder is a multiple of that integer.

Example:  12 is a multiple of 3, since 12 is divisible by 3.

Factors

The factors, or divisors, of a number are the positive integers that divide into that number without a remainder (or a remainder of 0).

Example:  The number 36 has nine positive factors: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

You can group these factors in pairs: 1 × 36 = 2 × 18 = 3 × 12 = 4 × 9 = 6 × 6

These rules can be synthesized into one: image.

The greatest common factor, or greatest common divisor, of a pair of numbers is the largest factor shared by the two numbers.

As with all number properties questions, Picking Numbers is a useful strategy to use with questions about factors and multiples. When Picking Numbers, keep in mind the following:

In-Format Question: Factors and Multiples on the GMAT

Now let’s use the Kaplan Method on a Problem Solving question dealing with factors and multiples:

If p, q, and r are positive integers such that q is a factor of r, and r is a multiple of p, which of the following must be an integer?

Step 1: Analyze the Question

You are given the information that q is a factor of r, so you know that r is a multiple of q. The question stem also says that r is a multiple of p. Thus, r is a multiple of both p and q. This question is very abstract, so Picking Numbers that are permissible will help you make quick work of the answer choices.

Step 2: State the Task

You need to pick numbers for p, q, and r and apply them to the answer choices. Pay Attention to the Right Detail: you are looking for the answer choice that is an integer. Eliminate any answer choices that do not yield an integer.

Step 3: Approach Strategically

Based on work in Step 1, you know that r is a multiple of both p and q. So if p = 2 and q = 3, then r = 12 meets the criteria stated in the question stem.

You can now substitute these values into the answer choices. Because this is a “which of the following” question, begin with (E).

Choice (E): image. The answer is an integer, so possibly correct.

Choice (D): image. Not an integer. Eliminate.

Choice (C): image. Not an integer. Eliminate.

Choice (B): image. Not an integer. Eliminate.

Choice (A): image. Not an integer. Eliminate.

Step 4: Confirm Your Answer

If all other four answer choices can give non-integers, (E) must be the answer. Note that if you had originally picked different numbers, you may have ended up with more than one answer choice that produced an integer value. Whenever this is the case, you would need to pick a new set of numbers and test only the answer choices that worked out the first time.

(E) also makes logical sense because it equals image, and according to the question stem, q and p are both factors of r, so image and image are both integers. Therefore, their sum must also be an integer.

TAKEAWAYS: FACTORS AND MULTIPLES

 

Practice Set: Factors and Multiples on the GMAT

  1. How many positive integers less than 50 are multiples of 4 but NOT multiples of 6?

    • 4

    • 6

    • 8

    • 10

    • 12

  2. If a certain number is divisible by 12 and 10, it is NOT necessarily divisible by which of the following?

    • 4

    • 6

    • 15

    • 20

    • 24

  3. What is the greatest positive integer x such that 3x is a factor of 910?

    • 5

    • 9

    • 10

    • 20

    • 30

REMAINDERS AND PRIMES

These two topics—remainders and prime numbers—are math concepts that you may not use a lot in day-to-day life. Although you may be rusty on these concepts, there’s no need to be intimidated. The most important thing for you to understand is what a remainder is and how it differs from a quotient. In most areas of life and work, you likely perform division using a calculator, which automatically calculates the decimal places. Remainders are tested on the GMAT, in part, because they are distinctly not what calculators give you.

Remainders

The remainder is what is left over in a division problem. A remainder is always smaller than the number you are dividing by.

Example:  17 divided by 3 is 5 with a remainder of 2.

The GMAT sometimes asks questions that require you to identify numbers that when divided by a given number produce a certain remainder. For example, “n is a number that when divided by 7 has a remainder of 2.” Most GMAT remainder problems like this one are best solved by the strategy of Picking Numbers.

Prime Numbers

A prime number is an integer greater than 1 that has only two positive factors, 1 and itself. The number 1 is not considered a prime. The number 2 is the first prime number and the only even prime. (Do you see why? Any other even number has 2 as a factor and, therefore, is not prime.) The GMAT expects test takers to recognize the prime numbers up to 50. They are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47.

Prime Factorization

The prime factorization of a number is the expression of the number as the product of its prime factors. No matter how you factor a number, its prime factors will always be the same.

Example:  36 = 6 × 6 = 2 × 3 × 2 × 3 = 22 × 32

Example:  480 = 48 × 10 = 8 × 6 × 2 × 5

  = 4 × 2 × 2 × 3 × 2 × 5

  = 2 × 2 × 2 × 2 × 3 × 2 × 5

  = 25 × 3 × 5

The easiest way to determine a number’s prime factorization is to figure out a pair of factors of the number and then determine their factors, continuing the process until you’re left with only prime numbers. Those primes will be the prime factorization.

Example:  Find the prime factorization of 1,050.

image

So the prime factorization of 1,050 is 2 × 3 × 52 × 7.

Prime factorization is one of the most valuable tools for the entire Quantitative section. Any question about multiples or factors is really, at its heart, a question about prime factors. Quickly jotting down the prime factorizations of the numbers in such questions can provide the key to the solution. Prime factorization is often the key to dealing with scary exponents as well. If a GMAT problem gives you 358, rewriting it as (5 × 7)8 = 58 × 78 is almost certainly the way to go.

Distinct Prime Factors

The GMAT asks not only about prime factors, but also about distinct prime factors. Distinct simply means “different.” Thus, 1,050 = 2 × 3 × 5 × 5 × 7, five numbers with 5 repeating once; it has five prime factors, but only four distinct ones.

Divisibility Tests

Several tests can quickly determine whether a given number is a multiple of 2, 3, 4, 5, 7, 8, 9, 10, and 11.

A number is divisible by 2 if its units digit is even.

138 is divisible by 2 because 8 is even.

177 is not divisible by 2 because 7 is not even.

A number is divisible by 3 if the sum of its digits is divisible by 3.

4,317 is divisible by 3 because 4 + 3 + 1 + 7 = 15 and 15 is divisible by 3.

32,872 is not divisible by 3 because 3 + 2 + 8 + 7 + 2 = 22 and 22 is not divisible by 3.

A number is divisible by 4 if its last two digits compose a two-digit number that is itself divisible by 4.

1,732 is divisible by 4 because 32 is divisible by 4.

1,746 is not divisible by 4 because 46 is not divisible by 4.

A number is divisible by 5 if its units digit is either a 5 or a 0.

26,985 is divisible by 5.

55,783 is not divisible by 5.

A number is divisible by 7 if the difference between its units digit multiplied by 2 and the rest of the number is a multiple of 7.

147 is divisible by 7 because 14 − 7(2) = 0, which is divisible by 7.

682 is not divisible by 7 because 68 − 2(2) = 64, which is not divisible by 7.

A number is divisible by 8 if its last three digits compose a three-digit number that is itself divisible by 8.

76,848 is divisible by 8 because 848 is divisible by 8.

65,102 is not divisible by 8 because 102 is not divisible by 8.

A number is divisible by 9 if the sum of its digits is divisible by 9.

16,956 is divisible by 9 because 1 + 6 + 9 + 5 + 6 = 27, and 27 is divisible by 9.

4,317 is not divisible by 9 because 4 + 3 + 1 + 7 = 15, and 15 is not divisible by 9.

A number is divisible by 10 if its units digit is zero.

67,890 is divisible by 10.

56,432 is not divisible by 10.

A number is divisible by 11 if the difference between the sum of its odd-placed digits and the sum of its even-placed digits is divisible by 11.

5,181 is divisible by 11 because (5 + 8) − (1 + 1) = 11, which is divisible by 11.

5,577 is a multiple of 11 because (5 + 7) − (5 + 7) = 0, which is divisible by 11.

827 is not divisible by 11 because (8 + 7) − 2 = 13, which is not divisible by 11.

Bonus rule for 11: If the digit in the tens place of a three-digit number is equal to the sum of the digits in that number’s hundreds and units places, then that number is divisible by 11.

Furthermore, the quotient of the number divided by 11 will be a two-digit number composed of the digits in the original number’s hundreds and units places. Divisibility by 11 shows up with surprising regularity in the GMAT’s tougher Quantitative problems.

341 is divisible by 11 because 3 + 1 = 4. Furthermore, 341 = 31 × 11.

792 is divisible by 11 because 7 + 2 = 9. Furthermore, 792 = 72 × 11.

This rule cannot be used to rule out divisibility, only to rule it in:

715 is divisible by 11, even though 7 + 5 ≠ 1 (715 = 65 × 11).

You can combine these rules above with factorization to figure out whether a number is divisible by other numbers.

To be divisible by 6, a number must pass the divisibility tests for both 2 and 3, since 6 = 2 × 3.

534 is divisible by 6 because 4 is divisible by 2 and 5 + 3 + 4 = 12, which is divisible by 3.

To be divisible by 44, a number must pass the divisibility tests for both 4 and 11, since 44 = 4 × 11.

1,848 is divisible by 44 because 48 is divisible by 4 and (1 + 4) − (8 + 8) = −11, which is divisible by 11.

Note that combining rules to test divisibility only works when the separate numbers you are testing do not have any factors in common (as is the case with 2 and 3 and with 4 and 11 above).

In-Format Question: Remainders and Primes on the GMAT

Now let’s use the Kaplan Method on a Problem Solving question dealing with remainders and primes:

If a and b are prime numbers, which of the following CANNOT be the value of ab?

Step 1: Analyze the Question

Given that a and b are prime numbers, you are looking for the one answer choice that cannot be the product of two prime numbers. Note though that when the question is testing prime numbers, remember to test the special case of the number 2, since 2 is both the smallest prime number and also the only even prime number.

Step 2: State the Task

Pick numbers that are permissible and systematically apply the numbers to the answer choices. Eliminate answer choices that cannot be the product of the two prime numbers.

Step 3: Approach Strategically

This is a “which of the following” question, so you can start with (E) and test the choices.

Choice (E): The product is 25, which can be created by multiplying 5 × 5. Because 5 is a prime number, 25 can be the product of two primes and, thus, cannot be the correct choice. Notice that the question stem requires only that a and b be prime numbers, not that they be distinct prime numbers. Eliminate.

Choice (D): The number 23 is prime; its only factors are 1 and 23. Because 1 is not a prime number, 23 cannot be the product of two prime numbers. Therefore, (D) is correct.

Step 4: Confirm Your Answer

Use Critical Thinking to check your work. Logically, it makes sense that the correct answer would be a prime number, since a and b, as prime numbers, must both be greater than 1, making it impossible for ab to be a prime number.

TAKEAWAYS: REMAINDERS AND PRIMES

 

Practice Set: Remainders and Primes on the GMAT

Answers and explanations at end of chapter

  1. If n is a positive integer greater than 16, is n a prime number?

(1)   n is odd.

(2)   The remainder when n is divided by 3 is 1, and the remainder when n is divided by 7 is 1.

  1. If 2 is the remainder when m is divided by 5, what is the remainder when 3m is divided by 5?

    • 0

    • 1

    • 2

    • 3

    • 4

  2. If a = 105 and a3 = 21 × 25 × 45 × b, what is the value of b?

    • 35

    • 42

    • 45

    • 49

    • 54

SEQUENCES

A sequence is an ordered list of numbers an−1, an, an+1, where n denotes where in the sequence the number is located. Sequence-based questions on the GMAT can look complicated, but once you learn to break down the notation, they’re not that tricky.

Example:  If n = 2, a2 is the 2nd number,

a2+1 = a3 is the 3rd number,

a2−1 = a1 is the 1st number, and so on.

Before you dive into statement and answer evaluation, make sure you understand what you are looking for. Use Paraphrasing to state the information in the question in your own words. Then apply Critical Thinking to identify the pattern in the sequence you’re given. (Since this is the GMAT, there will always be a discernible pattern to the sequence even if it isn’t transparent immediately.)

Once you have a thorough understanding of the question stem and the sequence, look at the statements and answers. Picking Numbers is a useful strategy if you’re not given concrete values for the terms in a sequence.

With a little patience and a strategic approach, you can attack even the most intimidating sequence questions effectively.

In-Format Question: Sequences on the GMAT

Now let’s use the Kaplan Method on a Problem Solving question dealing with sequences:

In an increasing sequence of 8 consecutive even integers, the sum of the first 4 integers is 268. What is the sum of all the integers in the sequence?

Step 1: Analyze the Question Stem

You are asked to determine the sum of all the integers in the sequence, and you are given the sum of the first four of eight integers. As discussed in the Statistics chapter of this book, half of the numbers in a consecutive sequence with an even number of terms must be greater than and half must be less than the mean.

Step 2: State the Task

Since you are given the sum of the first half of the sequence, you can use the average formula to solve this strategically.

Step 3: Approach Strategically

Since the sum of four terms is 268, the average is 67, which means the four even integers must be 64, 66, 68, and 70. This means that the next four must be 72, 74, 76, and 78. The sum of all eight is 568, and that’s your answer. You could also apply Critical Thinking and realize that since the numbers in the sequence are consecutive even integers, the first term in the second group will be 8 larger than the first term in the first group, the second term in the second group 8 larger than the second term in the first group, and so on. This means that the final four numbers add up to 268 + (8 + 8 + 8 + 8), or 300. Then 268 + 300 = 568. Choice (B) is correct.

Step 4: Confirm Your Answer

The numbers in the answer choices here are not very spread out. So estimation is not likely to be very helpful. A quick double-check, especially using the alternative method above, will confirm that you didn’t make any mistakes.

TAKEAWAY: SEQUENCES

 

Practice Set: Sequences on the GMAT

Answers and explanations at end of chapter

l, m, n, o, p

  1. An arithmetic sequence is a sequence in which each term after the first is equal to the sum of the preceding term and a constant. If the list of letters shown above is an arithmetic sequence, which of the following must also be an arithmetic sequence?

    1. 3l, 3m, 3n, 3o, 3p

    2. l2, m2, n2, o2, p2

    3. l − 5, m − 5, n − 5, o − 5, p − 5

    • I only

    • II only

    • III only

    • I and III

    • II and III

  2. In a certain sequence, the term an is given by the formula image for all n ≥ 3. If a1 = 2 and a2 = 6, what is the value of a6?

    • 12

    • 15

    • 25.5

    • 32.5

    • 40

  3. In the infinite sequence S, each term Sn after S2 is equal to the sum of the two terms Sn−1 and Sn−2. If S1 is 4, what is the value of S2?

    1. (1)   S3 = 7

    2. (2)   S4 = 10

Answer Key

Practice Set: Integers and Non-integers on the GMAT

  1. A

  2. B

  3. A

Practice Set: Odds and Evens on the GMAT

  1. D

  2. E

  3. E

Practice Set: Positives and Negatives on the GMAT

  1. C

  2. C

  3. C

Practice Set: Factors and Multiples on the GMAT

  1. C

  2. E

  3. D

Practice Set: Remainders and Primes on the GMAT

  1. E

  2. B

  3. D

Practice Set: Sequences on the GMAT

  1. D

  2. C

  3. D

Answers and Explanations

Practice Set: Integers and Non-integers on the GMAT

1.    (A)

Is z an integer?

(1)   image is an integer.

(2)   image is NOT an integer.

Step 1: Analyze the Question Stem

In this Yes/No question, sufficiency means demonstrating that z is either definitely an integer or definitely not an integer.

Step 2: Evaluate the Statements Using 12TEN

Statement (1) states that image is an integer, so z must be a multiple of 3. As every multiple of 3 is an integer, this statement is sufficient. Eliminate (B), (C), and (E).

Statement (2) states that image is not an integer. There are two ways for this to be true: either z is an odd integer, in which case it would still be an integer, or z is not an integer. There is no definite answer to the question, so this statement is insufficient. Eliminate (D).

Statement (1) alone is sufficient, so the correct answer is (A).

2.    (B)

If x and y are positive integers, is image an integer?

(1)   x + y = 30

(2)   image

Step 1: Analyze the Question Stem

This is a Yes/No question. We are told that x and y are positive integers. We need to figure out if there is sufficient information to determine whether image is an integer. The stem doesn’t provide much information, so let’s move on to the statements.

Step 2: Evaluate the Statements Using 12TEN

Statement (1) tells us that x + y = 30. If x = 15 and y = 15, then x + y = 15 + 15 = 30, so Statement (1) is true. In this case, image is an integer, so the answer to the question is “yes.” However, if x = 10 and y = 20, then x + y = 10 + 20 = 30, so Statement (1) is true. In this case, image is not an integer, so the answer to the question is “no.” Because more than one answer to the question is possible, Statement (1) is insufficient. We can eliminate (A) and (D).

Statement (2) tells us that image. Let’s find an equation equivalent to this equation that is simpler to work with. Multiplying both sides of this equation by y + 2, we get x − 4 = (y − 2)(y + 2). Expanding the right side of the equation, using FOIL, we have

image

Statement (2) can now be written as x = y2, and as y is positive, we can divide by y and get image = y. Because we know y is a positive integer, the answer to our question is “always yes”; image is always an integer. Answer choice (B) is correct.

We could also use Picking Numbers to evaluate this statement. If we pick some values for y, find the corresponding values for x, and then find the value of image, we will find it is always an integer.

For example, if y = 1, then x = y2 = 12 = 1, and image, which is an integer. If y = 2, then x = y2 = 22 = 4, and image, which is an integer. Statement (2) is sufficient to answer the question “yes.” (B) is correct.

3.    (A)

If d is a positive integer, is image an integer?

(1)   image is an integer.

(2)   image is NOT an integer.

Step 1: Analyze the Question Stem

This is a Yes/No question. We are told that d is a positive integer. We need to determine if there is enough information to answer the question “Is image an integer?” We can paraphrase the question stem to say, “Is d a perfect square?” Let’s look at the statements.

Step 2: Evaluate the Statements Using 12TEN

Because number properties are involved in our statements, it may be easiest to use Picking Numbers.

Statement (1) tells us that image is an integer. This means that 9d is a perfect square. So some permissible values for d are 1, 4, and 9, which all have integer square roots. Statement (1) is sufficient to answer the question “yes.” Eliminate (B), (C), and (E).

To evaluate Statement (1) algebraically, we can use our rules for radicals to simplify it. We know that image is an integer, so image is an integer and image is an integer. Our integer result must be a multiple of 3, and image must be a factor of that number and itself an integer. We can also analyze this statement by paraphrasing it to say, “9d is a perfect square.” Since 9 itself is a perfect square, d alone must also be a perfect square, and the answer to the question stem is “always yes.”

Statement (2) tells us image is not an integer. Again Picking Numbers to evaluate this statement, we can choose d = 1. We know that image is not an integer, so this is an acceptable value of d, and image = 1, an integer, so the answer to our question is “yes.” But if we choose d = 2, image. This is not an integer, so 2 is an acceptable value for d and image is not an integer, and the answer to our question is “no.” Because more than one answer to the question is possible, Statement (2) is insufficient. Eliminate (D), and our correct answer is (A).

Practice Set: Odds and Evens on the GMAT

4.    (D)

If z is an integer, is z even?

(1)   image, where m is an integer.

(2)   z3 is even.

Step 1: Analyze the Question Stem

In this Yes/No question, we are told that z is an integer. Sufficiency means showing that z is either definitely even or definitely odd.

Step 2: Evaluate the Statements Using 12TEN

Statement (1) provides an equation involving the integer m. Let’s investigate by simplifying the equation:

image

So z is 6m − 4. Picking Numbers for m can help us determine whether z needs to be even. If m = 0 (an even number), then z = −4. If m = 1 (an odd number), then z = 6 − 4 = 2. Whether m is odd or even, z ends up being even, so this statement is sufficient. Eliminate (B), (C), and (E).

Statement (2) tells us that z3 is even. If we know our odds and evens rules, we know this means that z is even, but if we don’t remember that rule, or if we want to double-check, we can use Picking Numbers.

Let’s pick some different numbers for z: 13 = 1, 23 = 8, 33 = 27, 43 = 64. Only even numbers raised to the third power end up even, so this statement is sufficient. Eliminate (A).

Each statement alone is sufficient to answer the question, so (D) is the correct answer.

5.    (E)

The set S contains n integers. Is the sum of all the elements of set S odd?

(1)   All the elements of S are prime numbers.

(2)   n = 2

Step 1: Analyze the Question Stem

This is a Yes/No question. We want to determine whether the sum of all the elements of set S is odd. The only information that the question stem gives us is that S contains n integers, so let’s look at the statements.

Step 2: Evaluate the Statements Using 12TEN

Statement (1) says that all the elements of S are prime numbers. We should remember that there is one even prime number, the integer 2. All other prime numbers are odd.

If S = {2, 3}, then the sum of all the elements of S is 2 + 3 = 5, which is odd. In this case, the answer to the question is “yes.” If S = {3, 5}, then the sum of all the elements of S is 3 + 5 = 8, which is even. In this case, the answer to the question is “no.” Because there is more than one possible answer to the question, Statement (1) is insufficient. We can eliminate (A) and (D).

Statement (2) says that n = 2; that is, there are two integers in set S. When we considered Statement (1), we let S = {2, 3}. If S = {2, 3}, then Statement (2) is true, and the answer to the question is “yes” because 2 + 3 is odd. When we considered Statement (1), we also let S = {3, 5}. If S = {3, 5}, then Statement (2) is true, and the answer to the question is “no” because 3 + 5 is not odd.

Because there is more than one possible answer to the question, Statement (2) is insufficient. We can eliminate (B).

Now let’s take the statements together. If S = {2, 3}, then both statements are true, and the answer to the question is “yes.” If S = {3, 5}, then both statements are true, and the answer to the question is “no.” Because there is more than one possible answer to the question when both statements are taken together, the two statements taken together are insufficient. Choice (E) is correct.

6.    (E)

If negative integers k and p are NOT both even, which of the following must be odd?

Step 1: Analyze the Question

For this abstract number properties question, we can either apply the rules for odd and even numbers directly or simply pick some numbers to solve the question.

Step 2: State the Task

We must determine which answer choice must always be odd or, in other words, eliminate any answer choices that can be even.

Step 3: Approach Strategically

For some number properties questions, using rules (if you are certain of them) is faster than Picking Numbers. In this question, the condition that k and p are negative and are “not both even” complicates Picking Numbers but not applying rules.

Since we have rules for odd and even numbers, we can apply them directly to the answer choices. We start with (E), since this is a “which of the following” question.

(E): Because 2 times any value is even, 2(k + p) will always be even. Subtracting one from an even number will always result in an odd number. Therefore, (E) is always odd and must be the correct answer.

Step 4: Confirm Your Answer

You can confirm your answer by noting that (A) through (D) could be even, judging by odd and even rules. (B) is always even, for example. (A) is odd when k and p are both odd, but the question stem allows for the possibility that one of them is even, and in such a case kp is even.

Practice Set: Positives and Negatives on the GMAT

7.    (C)

If x − y = 8, which of the following must be true?

  1. Both x and y are positive.

  2. If x is positive, y must be positive.

  3. If x is negative, y must be negative.

Step 1: Analyze the Question

We know that x − y = 8. As this is a Roman numeral question with variables, expect to use Picking Numbers and don’t forget to check the Roman numeral statements in the most efficient order. In this question, we will begin with Roman numeral II, as it shows up the most in the answer choices.

Step 2: State the Task

Determine which of the Roman numeral statements are true.

Step 3: Approach Strategically

Roman numeral II: If x is positive, y must be positive. To test whether this statement needs to be true, let’s try to pick some numbers for x and y where the result is false. The key is to pick numbers such that the condition is true but the result is false. If x = 6 and y = −2, then x − y does equal 8. The condition is true yet we were able to make the result false, so Roman numeral II does not need to be true. Eliminate (B), (D), and (E).

Of the remaining statements, Roman numeral I (Both x and y are positive) is the easier one to test because the numbers we just picked for x and y can be used to invalidate this statement as well. Eliminate (A).

Therefore, the correct answer is (C).

Step 4: Confirm Your Answer

Confirm the answer by checking Roman numeral III: If x is negative, y must be negative. In the equation x − y = 8, if x is negative and y is positive, then the result would simply get more negative. The only way to arrive at a positive answer when x is negative is for y to also be negative. Choice (C) is confirmed.

8.    (C)

If x and y are integers such that x > y, which of the following CANNOT be a positive integer?

  1. y(y − x)

  2. y(x − y)

  3. image

Step 1: Analyze the Question

As with most number properties questions, the most efficient strategy to solving this question will be to pick some numbers for the variables that are manageable and permissible. Also, since this is a Roman numeral question, we should determine which Roman numeral appears in the answer choices most frequently and assess that one first. In this question, the most frequently appearing Roman numeral is II.

Step 2: State the Task

Pick numbers for the variables x and y and eliminate the statements that can yield a positive integer.

Step 3: Approach Strategically

Roman numeral II: y(x − y). Since x > y, we can pick x = 2 and y = 1. Then y(x − y) = (1)(2 − 1) = (1)(1) = 1. So if y is also positive, then y(x − y) is a positive times a positive, which is positive. Because Roman numeral II can be a positive integer, this option will not be part of the correct answer. We can eliminate (B), (D), and (E) because these choices contain Roman numeral II.

Roman numeral I: y(y − x). Let’s test the negative case. If x = −3 and y = −4, then y(y − x) = (−4)[−4 − (−3)] = (−4)(−4 + 3) = (−4)(−1) = 4. Because Roman numeral I can be a positive integer, this option will not be part of the correct answer. We can eliminate (A) (and (D), if you didn’t do so already).

We have now eliminated all four incorrect answer choices, so the correct answer must be (C).

Step 4: Confirm Your Answer

Just to be sure, let’s check (C). If we use Picking Numbers, we will always find that image, which is negative. This is because the numerator x − y is the negative of the denominator y − x. So image. We canceled a factor of y − x from the numerator and denominator to obtain −1. Because Roman numeral III is always −1, it cannot ever be a positive integer.

9.    (C)

Is x > y?

(1)   9x = 4y

(2)   x > −y

Step 1: Analyze the Question Stem

This is a Yes/No question. The stem does not give us much information, so let’s go directly to the statements, looking for information about the relationship between x and y.

Step 2: Evaluate the Statements Using 12TEN

Let’s rewrite Statement (1) as image = y. Now we can use Picking Numbers. If x = 4, then y = 9, so x is not greater than y in this case. But x could also be negative. If x = −4, then y = −9, and now x is greater than y. Different answers to the question are possible, so Statement (1) is insufficient. We can eliminate (A) and (D).

If x is any positive number, then Statement (2) will be true for any non-negative value of y, which means that x may or may not be greater than y. For example, if x = 5, y could be 1 or y could be 10. Either value of y works, but in one case x is greater than y and in the other x is less than y. So Statement (2) is also insufficient. We can eliminate (B).

To evaluate the two statements together, we can combine the equation in Statement (1) with the inequality from Statement (2). Because image = y, we can replace y in the Statement (2) inequality with image. This gives us x > −image, or x + image > 0. Rewriting x as image, we have image + image > 0, or image > 0, so x must be positive.

Now we return to image = y. If x must be greater than zero, then any value we pick for x will make y larger than x. For example, if x = 4, y = 9. So x can never be greater than y. The answer to the question stem is “always no,” meaning that the two statements together are sufficient. The correct answer is (C).

Practice Set: Factors and Multiples on the GMAT

10.   (C)

How many positive integers less than 50 are multiples of 4 but NOT multiples of 6?

Step 1: Analyze the Question

This question involves multiples of 4 and multiples of 6. When dealing with such problems, remember that prime factorization can often provide a much faster solution than doing things the hard way.

Step 2: State the Task

Find the number of multiples of 4 less than 50 that aren’t also multiples of 6.

Step 3: Approach Strategically

The prime factorization of 4 and 6 is 2 × 2 and 2 × 3, respectively. This shows us that we need a minimum of two 2s and one 3 to “make” either a 4 or a 6 (not both at once), so a number that is a multiple of both 4 and 6 is a multiple of 2 × 2 × 3 = 12. To find the number of multiples of 4 that aren’t also multiples of 6, we should find the number of multiples of 4 under 50 and subtract the number of multiples of 12 under 50.

Dividing 4 into 50 yields 12 and a remainder. Dividing 12 into 50 yields 4 and a remainder. That means there are 12 multiples of 4 that are less than 50 and 4 multiples of 12 that are less than 50. (Note: had either one not yielded a remainder, we would need to subtract 1 from that count since these are “less than 50” and not “50 or less.”) Therefore, there are 12 − 4 = 8 multiples of 4 under 50 that aren’t also a multiple of 6. Choice (C) is correct.

Step 4: Confirm Your Answer

To confirm the answer, write out all of the multiples of 4 under 50, then cross out any that are multiples of 6. The multiples of 4 under 50 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, and 48. Of these, 12, 24, 36, and 48 are multiples of 6. Crossing those out leaves eight numbers, so (C) is confirmed.

11.   (E)

If a certain number is divisible by 12 and 10, it is NOT necessarily divisible by which of the following?

Step 1: Analyze the Question Stem

We are asked to consider common multiples of 10 and 12 and determine which answer choice may not also be a factor of their common multiples.

Step 2: State the Task

To determine the least common multiple of two numbers, we use the prime factorization of those numbers.

Step 3: Approach Strategically

Let’s begin by finding the least common multiple of 10 and 12. We know 10 has two prime factors: 2 and 5. And 12 has three prime factors: 2, 2, and 3. Combining those, we find that the least common multiple of 10 and 12 is 2 × 2 × 3 × 5 = 60. Likewise, 120, 180, 240, etc. will also be multiples of 10 and 12. Checking our answer choices, we see that 4, 6, 15, and 20 are all factors of 60 but (E), 24, is not. (E) is the correct choice.

Step 4: Confirm Your Answer

The testmaker sets several traps here. The easiest trap to fall into is to miss the NOT in the question stem and pick an answer that is a factor of 60. On Test Day, you never want to lose a point for which you did all of the math correctly and then answered the wrong question. Another potential pitfall is to pick just one multiple of 10 and 12—say 240—and dismiss (E) because 24 is a factor of 240. The wording in the question stem says that the correct answer is not necessarily a factor of a multiple of both 10 and 12. This is one reason that finding the least common multiple is a good rule of thumb in questions like this one.

12.   (D)

What is the greatest positive integer x such that 3x is a factor of 910?

Step 1: Analyze the Question

This question asks us to think critically about exponents. It’s crucial that we note that the base number of the first term—3—is a factor of the base number in the second—9.

Step 2: State the Task

Our task is to find the largest possible exponent x that will make 3x a factor of 910. We must remember that every number is a factor of itself.

Step 3: Approach Strategically

Good Critical Thinking saves us a lot of math and scratchwork on this question. Since 9 is 3 × 3 = 32, we can simply conceive of 910 as (32)10. That’s the same as 320. And, since every number is a factor of itself, 20 is the largest exponent that will work in this case. Choice (D) is correct.

Step 4: Confirm Your Answer

Avoiding unnecessary work is the key to avoiding mistakes on questions like this one. It’s not going to be very helpful to try writing out 3 × 3 × 3 × 3 … to test answer choices. Good Critical Thinking is all you need.

Practice Set: Remainders and Primes on the GMAT

13.   (E)

If n is a positive integer greater than 16, is n a prime number?

(1)   n is odd.

(2)   The remainder when n is divided by 3 is 1, and the remainder when n is divided by 7 is 1.

Step 1: Analyze the Question Stem

In this Yes/No question, we are told that n is a positive integer greater than 16. Sufficiency means showing that n is either definitely prime or definitely not prime.

Step 2: Evaluate the Statements Using 12TEN

Statement (1) notes that n is odd. While prime numbers above 16 are certainly all odd, there are some odd numbers—for example, odd multiples of 3 or 5—that are not prime. This statement is insufficient. Eliminate (A) and (D).

Statement (2) notes that n divided by 3 and n divided by 7 both yield a remainder of 1. As 3 and 7 do not have any factors in common, this is equivalent to saying that n divided by 3 × 7 = 21 yields a remainder of 1. That means n can be 21 + 1 = 22, 2(21) + 1 = 43, 3(21) + 1 = 64, or 4(21) + 1 = 85, just to name the first four possibilities. Of these four, 43 is prime but the other three are not, so this statement is insufficient. Eliminate (B).

Combining the statements reveals that n is odd and 1 more than a multiple of 21. This too is insufficient because the first two possible values of n are 43 and 85 and 43 is prime while 85 is not. Eliminate (C).

The statements, even when combined, are insufficient, so the correct answer is (E).

14.   (B)

If 2 is the remainder when m is divided by 5, what is the remainder when 3m is divided by 5?

Step 1: Analyze the Question

This question tests our ability to think critically about the characteristics of remainders in division. We are told that some number, m, has a remainder of 2 when divided by 5.

Step 2: State the Task

We can use our knowledge of number properties to take a particularly strategic approach to this problem. The key will be to pick simple, permissible numbers and apply them to the problem in the question stem.

Step 3: Approach Strategically

Ask yourself what numbers would be permissible for m. Since m has a remainder of 2 when divided by 5, m could be any number 2 greater than a multiple of 5. The simplest number to substitute for m is 7. We know that 5 goes into 7 one time with a remainder of 2. Now, apply 7 to the rest of the question stem: 3m divided by 5. Well, 3 × 7 = 21, and 21 divided by 5 would leave a remainder of 1. That’s (B). Picking Numbers is the most efficient approach to this common GMAT question type.

Step 4: Confirm Your Answer

To double-check your work, you could test any other permissible number for m: 12, 17, 22, etc. If you tried 12, you would find that 3 × 12 = 36 and 36 divided by 5 leaves a remainder of 1. This confirms that (B) is the correct choice.

15.   (D)

If a = 105 and a3 = 21 × 25 × 45 × b, what is the value of b?

Step 1: Analyze the Question

We are told that a = 105 and that a3 = 21 × 25 × 45 × b. We want to determine the value of b. This question is a great candidate for prime factorization, since we need to factor a large product to determine the unknown factor b.

Step 2: State the Task

Determine the prime factors of each term in the product and then cancel out all of the common terms. Those that remain will be the prime factors of b.

Step 3: Approach Strategically

Let’s find the prime factorization of each term:

image

Now we can plug the prime factors as products into the equation from the question stem and solve for b:

image

Choice (D) is the correct answer.

Step 4: Confirm Your Answer

When using prime factorization, be careful to list all of the prime factors, including ones that repeat, such as 3 × 3 or 5 × 5.

Practice Set: Sequences on the GMAT

16.   (D)

l, m, n, o, p

An arithmetic sequence is a sequence in which each term after the first is equal to the sum of the preceding term and a constant. If the list of letters shown above is an arithmetic sequence, which of the following must also be an arithmetic sequence?

  1. 3l, 3m, 3n, 3o, 3p

  2. l2, m2, n2, o2, p2

  3. l − 5, m − 5, n − 5, o − 5, p − 5

Step 1: Analyze the Question

We are provided with the definition of an arithmetic sequence and told that l, m, n, o, p is one such sequence. As this is a Roman numeral question with variables, expect to use Picking Numbers and don’t forget to check the Roman numeral statements in the most efficient order. In this question, we will want to begin with Roman numeral III, as it shows up the most in the answer choices.

Step 2: State the Task

Determine which of the Roman numeral statements are true.

Step 3: Approach Strategically

Roman numeral III: To test this statement, let’s say our original arithmetic sequence is 6, 9, 12, 15, 18 (adding a constant of 3 each time). Roman numeral III would have us subtract 5 from each term, which results in the sequence 1, 4, 7, 10, 13. This is still an arithmetic sequence as each term is exactly 3 more than its preceding term, so Roman numeral III is a true statement. Eliminate (A) and (B).

Of the remaining statements, Roman numeral I is easier to test. Using the same original sequence we had above, Roman numeral I would have us multiply each term by 3. Doing so results in the sequence 18, 27, 36, 45, 54. This is still an arithmetic sequence as each term is exactly 9 more than its preceding term, so Roman numeral I is a true statement. Eliminate (C) and (E).

Therefore, the correct answer is (D).

Step 4: Confirm Your Answer

To confirm the answer, let’s test Roman numeral II: Using the same original sequence we picked above, Roman numeral II would have us square each term. The larger numbers can make this time-consuming, but just squaring the first three terms (36, 81, 144) should be enough to show us that Roman numeral II does not result in an arithmetic sequence. Choice (D) is confirmed.

17.   (C)

In a certain sequence, the term an is given by the formula image for all n ≥ 3. If a1 = 2 and a2 = 6, what is the value of a6?

Step 1: Analyze the Question

We are given a formula for the sequence and the values for the first two terms of the sequence.

Step 2: State the Task

The problem gives us a formula to use in calculating all the relevant numbers we need in the problem. It simply requires us to plug numbers into the series until we get the sixth term.

Step 3: Approach Strategically

Plugging the information given into the formula, we would get image.

Plugging in image.

Plugging in image.

Finally, image.

That’s choice (C).

Step 4: Confirm Your Answer

The first two answers here are the results for the fourth and fifth numbers in the sequence. Checking your scratchwork to make sure you’ve calculated through the sixth number will help you avoid these wrong answer traps.

18.   (D)

In the infinite sequence S, each term Sn after S2 is equal to the sum of the two terms Sn−1 and Sn−2. If S1 is 4, what is the value of S2?

(1)   S3 = 7

(2)   S4 = 10

Step 1: Analyze the Question Stem

This is a Value question. We must determine what information is needed to solve directly for the term S2. The question stem says that if Sn is the nth term of the sequence, then for n > 2, we have Sn = Sn−1 + Sn−2. That is, after the second term in the sequence, the next term is equal to the sum of the two previous terms. For example, the third term is equal to the sum of the first and second terms.

We also know that S1 is 4. Because S3 = S2 + S1, S3 = S2 + 4. So, we have one linear equation with two variables, S3 and S2. To find the value of S2, we need more information that will lead to a single possible value for S2.

We could find the value of S2 if we were given the value of S3. We could also find the value of S2 if we were given another linear equation with the variables S3 and S2 that is different from the first equation. Let’s look at the statements.

Step 2: Evaluate the Statements Using 12TEN

Statement (1) says that S3 = 7. Because S3 = S2 + 4, we have the equation 7 = S2 + 4. From this equation, we can find the single possible value of S2. Statement (1) is sufficient. We can eliminate (B), (C), and (E).

Statement (2) says that S4 = 10. Because S4 = S3 + S2, we have the equation S3 + S2 = 10. This is another linear equation containing the terms S2 and S3. Because the two equations are distinct, we have enough information to determine the values for both S3 and S2. Statement (2) is sufficient. Choice (D) is correct.

GMAT BY THE NUMBERS: NUMBER PROPERTIES

Now that you’ve learned how to approach number properties questions on the GMAT, let’s add one more dimension to your understanding of how they work.

Take a moment to try the following question. The next page features performance data from thousands of people who have studied with Kaplan over the decades. Through analyzing this data, we will show you how to approach questions like this one most effectively and how to avoid similarly tempting wrong answer choice types on Test Day.

If a and b are positive integers, is 3a2b divisible by 60?

(1)   a is divisible by 10.

(2)   b is divisible by 18.
 

Explanation

The question asks whether 3a2b is divisible by 60. In other words, it asks whether image is an integer. The 3 in the numerator cancels out a factor of 3 from the denominator, so you only need to know whether image is an integer. For this expression to be an integer, the remaining factors in the denominator will have to be canceled out by the numerator. 20 = 2 × 2 × 5, so you’re really asked whether a2b contains two 2s and one 5 among its factors.

Statement (1) says that a is divisible by 10. Since 10 = 2 × 5, you now know that a contains at least one 2 and at least one 5 among its factors. However, the expression you are asked about contains not a but rather a2. If a has at least one 2 and at least one 5 among its factors, then a2 must have at least two 2s and at least two 5s. It doesn’t matter what factors b contains, since a2 alone provides all the factors you need. Statement (1) is therefore sufficient.

Since 18 = 2 × 3 × 3, Statement (2) tells you that b has at least one 2 and at least two 3s among its factors. By itself, b does not guarantee two factors of 2, let alone any factor of 5. Therefore, Statement (2) is insufficient.

Since Statement (1) alone is sufficient but Statement (2) is not, (A) is correct.

QUESTION STATISTICS
49% of test takers choose (A)
3% of test takers choose (B)
37% of test takers choose (C)
3% of test takers choose (D)
8% of test takers choose (E)
Sample size = 5,507

As the question statistics reveal, many test takers automatically assume that they need information about every variable in order to answer a question. Knowing this, the testmaker often writes statements that provide sufficient information despite not telling you about each individual variable. Don’t automatically think that you need information on all the variables—analyze the question stem carefully to see what you really need, and you’ll put yourself well ahead of your competition.

More GMAT by the Numbers …

To see more questions with answer choice statistics, be sure to review the full-length CATs in your online resources.