Integers and Non-Integers

Learning Objectives

In this section, you will learn how to: 

• Define integer and describe the number properties pertaining to integers that are commonly tested on the GMAT
• Distinguish different types of integers and non-integers
• Apply the Kaplan Method for Data Sufficiency to questions involving integers and non-integers

Integers are a particularly useful number properties category for the GMAT test makers, 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.

Types of Integers and Non-Integers

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, − ).

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:

• When an integer is added to, subtracted from, or multiplied by another integer, the result is an integer.
• An integer divided by an integer may or may not result in an integer.

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:

1. 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

• The term integer refers to positive whole numbers, zero, and negative whole numbers.
• When an integer is added to, subtracted from, or multiplied by another integer, the result is an integer. An integer divided by an integer may or may not result in an integer.
• When integer is a central word in a question, you have a number properties question. Take note of whether you are using rules or picking numbers.

Practice Set: Integers and Non-Integers on the GMAT

1. If each of d, q, and r is a positive integer such that dq + r = 3, what is the value of d?
   1. The number  is less than .
   2. The integer  is less than 2.
2. If f and g are positive integers, is   an integer?
   1. g = 5f − 4
   2. 
3. If n is a positive integer, is   an integer?
   1.  is an integer.
   2.  is NOT an integer.