Say that you are told some unknown positive number x is divisible by 6. How can you represent this on paper? There are many ways, depending on the problem. You could say that you know that x is a multiple of 6, or you could say that x = 6 × an integer. You could also represent the information with a factor tree. Careful though—although you’ve had a lot of practice drawing factor trees, there is one important difference now that you’re dealing with an unknown number. You know that x is divisible by 6, but x may be divisible by other numbers as well. You have to treat what they have told you as incomplete information, and remind yourselves there are other things about x you don’t know. To represent that on the page, your factor tree could look like this:
Now the question becomes—what else do you know about x? If a question on the GRE told you that x is divisible by 6, what could you definitely say about x? Take a look at these three statements, and for each statement, decide whether it must be true, whether it could be true, or whether it cannot be true.
Deal with each statement one at a time, beginning with Statement I—x is divisible by 3. One approach to take here is to think about the multiples of 6. If x is divisible by 6, then you know that x is a multiple of 6. List out the first several multiples of 6, and see if they’re divisible by 3.
At this point, you can be fairly certain that x is divisible by 3. In fact, listing out possible values of a variable is often a great way to begin answering a question in which you don’t know the value of the number you are asked about.
But can you do better than say you’re fairly certain x is divisible by 3? Is there a way to definitively say x must be divisible by 3? As it turns out, there is. Look at the factor tree for x again:
Remember, the ultimate purpose of the factor tree is to break numbers down into their fundamental building blocks: prime numbers. Now that the factor tree is broken down as far as it will go, you can apply the factor foundation rule. Thus, x is divisible by 6, and 6 is divisible by 3, so you can say definitively that x must be divisible by 3.
In fact, questions like this one are the reason so much time was spent discussing the factor foundation rule and the connection between prime factors and divisibility. Prime factors provide the foundation for a way to make definite statements about divisibility. With that in mind, look at Statement II.
Statement II says x is even. This question is about divisibility, so the question becomes, what is the connection between divisibility and a number being even? Remember, an important part of this test is the ability to make inferences based on the given information.
What’s the connection? Well, being even means being divisible by 2. So if you know that x is divisible by 2, then you can guarantee that x is even. Look at the factor tree:
You can once again make use of the factor foundation rule—6 is divisible by 2, so x must be divisible by 2 as well. And if x is divisible by 2, then x must be even as well.
That just leaves the final statement. Statement III says x is divisible by 12. Look at this question from the perspective of factor trees, and compare the factor tree of x with the factor tree of 12:
What would you have to know about x to guarantee that it is divisible by 12? Well, when 12 is broken down all the way, 12 is 2 × 2 × 3. Thus, 12’s building blocks are two 2’s and a 3. For x to be divisible by 12, it would have to also have two 2’s and one 3 among its prime factors. In other words, for x to be divisible by 12, it has to be divisible by everything that 12 is divisible by.
You need x to be divisible by two 2’s and one 3 in order to say it must be divisible by 12. But looking at your factor tree, there is only one 2 and only one 3. Because there is only one 2, you can’t say that x must be divisible by 12. But then the question becomes, could x be divisible by 12? Think about the question for a second, and then keep reading.
The key to this question is the question mark that you put on x’s factor tree. That question mark should remind you that you don’t know everything about x. Thus, x could have other prime factors. What if one of those unknown factors was another 2? Then the tree would look like this:
So if one of those unknown factors were a 2, then x would be divisible by 12. The key here is that you have no way of knowing for sure whether there is a 2. Thus, x may be divisible by 12, it may not. In other words, x could be divisible by 12.
To confirm this, go back to the multiples of 6. You still know that x must be a multiple of 6, so start by listing out the first several multiples and see whether they are divisible by 12.
Once again, some of the possible values of x are divisible by 12, and some aren’t. The best you can say is that x could be divisible by 12.
For these statements, the following is true: x is divisible by 24. For each statement, say whether it must be true, could be true, or cannot be true.
x is divisible by 6
x is divisible by 9
x is divisible by 8
Consider the following question, which has an additional twist this time. Once again, there will be three statements. Decide whether each statement must be true, could be true, or cannot be true. Answer this question on your own, then explore each statement one at a time on the next page.
x is divisible by 3 and by 10.
Before diving into the statements, spend a moment to organize the information the question has given you. You know that x is divisible by 3 and by 10, so you can create two factor trees to represent this information:
Now that you have your trees, get started with statement I. Statement I says that x is divisible by 2. The way to determine whether this statement is true should be fairly familiar by now—use the factor foundation rule. First of all, your factor trees aren’t quite finished. Factor trees should always be broken down all the way until every branch ends in a prime number. Really, your factor trees should look like this:
Now you are ready to decide whether statement I is true. Because x is divisible by 10, and 10 is divisible by 2, therefore x is divisible by 2. Statement I must be true.
That brings you to statement II. This statement is a little more difficult. It also requires you to take another look at your factor trees. You have two separate trees, but they’re giving you information about the same variable—x. Neither tree gives you complete information about x, but you do know a couple of things with absolute certainty. From the first tree, you know that x is divisible by 3, and from the second tree you know that x is divisible by 10—which really means you know that x is divisible by 2 and by 5. You can actually combine those two pieces of information and represent them on one factor tree, which would look like this:
Now you know three prime factors of x: 2, 3, and 5. Return to the statement. Statement II says that x is divisible by 15. What do you need to know to say that x must be divisible by 15? If you can guarantee that x has all the prime factors that 15 has, then you can guarantee that x is divisible by 15.
The number 15 breaks down into the prime factors 3 and 5. So to guarantee that x is divisible by 15, you need to know it’s divisible by 3 and by 5. Looking back up at your factor tree, notice that x has both a 3 and a 5, which means that x is divisible by 15. Therefore, statement II must be true.
You can also look at this question more visually. Remember, prime factors are like building blocks—x is divisible by any combination of these prime factors. You can combine the prime factors in a number of different ways, as shown here:
Each of these factor trees can tell you different factors of x. But what’s really important is what they have in common. No matter what way you combine the prime factors, each tree ultimately leads to 2 × 3 × 5, which equals 30. So you know that x is divisible by 30. And if x is divisible by 30, it is also divisible by everything 30 is divisible by. You know how to identify every number 30 is divisible by—use a factor pair table. The factor pair table of 30 looks like this:
| Small | Large |
| 1 | 30 |
| 2 | 15 |
| 3 | 10 |
| 5 | 6 |
Again, Statement II says that x is divisible by 15. Because x is divisible by 30, and 30 is divisible by 15, then x must be divisible by 15.
That brings you to Statement III. Statement III says that x is divisible by 45. What do you need to know to say that x must be divisible by 45? Build a factor tree of 45, which looks like this:
The number 45 is divisible by 3, 3, and 5. For x to be divisible by 45, you need to know that it has all the same prime factors. Does it?
The factorization of 45 has one 5 and two 3’s. Although x has a 5, you only know that x has one 3. That means that you can’t say for sure that x is divisible by 45. However, x could be divisible by 45, because you don’t know what the question mark contains. If it contains a 3, then x is divisible by 45. If it doesn’t contain a 3, then x is not divisible by 45. Without more information, you can’t say for sure either way. So statement III could be true.
Now it’s time to recap what’s been covered in this chapter. When dealing with questions about divisibility, you need a quick, accurate way to identify all the factors of a number. A factor pair table provides a reliable way to make sure you find every factor of a number.
Prime factors provide essential information about a number or variable. They are the fundamental building blocks of every number. In order for a number or variable to be divisible by another number, it must “contain” all the same prime factors that the other number contains. In the last example, you could definitely say that x was divisible by 15, because x contained one 3 and one 5. But you could not say for sure that it was divisible by 45, because 45 has one 5 and two 3’s, but x is only known to contain one 5 and one 3.
For these statements, the following is true: x is divisible by 28 and by 15. For each statement, say whether it must be true, could be true, or cannot be true.
x is divisible by 14.
x is divisible by 20.
x is divisible by 24.