Welcome back to our series on number properties. Last time we got familiar with prime factors as the substance of divisibility. In this article, we will extend this understanding by highlighting logical rules that sometimes come into play on GMAT number properties questions. Here is a list of related rules, to be followed by an explanation and some official practice problems:

## Prime factors: Logical rules on the GMAT

• General: If integer x is a factor of integer n, every factor of x is also a factor of n.
• Specific: If integer y is a factor of integer x and x is a factor of integer n, y is a factor of n.
• General: If integer x has a factor by which integer n is not divisible, x is not a factor of n.
• Specific: If integer y is a factor of integer x but not of integer n, x is not a factor of n.

For those familiar with formal logic, the second general/specific rule pair is the contrapositive of the first pair. For the rest of us logical laypeople, here’s what that means: x being a factor of n would guarantee the same status for each factor of x. So if we find any factor of x that is not also a factor of n, then x must not be a factor of n.

To generalize this principle: if a guaranteed effect of a particular cause does not occur, then the cause must not have occurred.

• If it had rained, the sidewalk would be wet.
The sidewalk is not wet, so it must not have rained.
• If my dog sees the squirrel, he will bark.
My dog is not barking, so he must not see the squirrel.

• If either team scores a goal, the game will end.
The game is still being played, so neither team has scored a goal.
• If x is a factor of n, every factor of x is also a factor of n.
x has a factor, y, that is not a factor of n, so x is not a factor of n.

The rule is simple enough, but it can have powerful applications.

### Data sufficiency problem

For starters, here’s a relatively easy data sufficiency problem:

Is the integer p divisible by 5?

1. p is divisible by 10.
2. p is not divisible by 15.

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

(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are not sufficient.

Let’s start with statement 1. Since p is divisible by 10, p is also divisible by every factor of 10. 5 is one of these factors, so p is divisible by 5. Statement 1 is sufficient.

Statement 2 is negative and tells us that p is NOT divisible by 15. But to conclude from this fact that p is not divisible by 5 would be a misapplication of our logic. Since p is not divisible by 15, p is not divisible by any multiple of 15. But p may be divisible by a given factor of 15, like 5. Therefore statement 2 is insufficient, and the correct answer is A.

### Here’s another interesting data sufficiency problem:

If m is an integer greater than 1, is m an even integer?

1. 32 is a factor of m.
2. m is a factor of 32.

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

(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are not sufficient.

Statement 1 is fairly easy to deal with and highlights a rule about evens and odds that will be fully addressed in a later article: if integer m has an even factor (like 32), then integer m is even. Instead of saying, “32 is a factor of m,” we could say, “m is a multiple of 32.” And every multiple of 32 will, of course, be even.

Statement 2 requires a bit more thought. The relationship from statement 1 is reversed, and we are told instead that m is a factor of 32. This is not immediately sufficient in the same way statement 1 is. Even integers cannot have odd multiples, but they can have odd factors. The data that integer m is a factor of an even number is not in itself sufficient to determine the evenness/oddness of integer m. But this particular even number, 32, has only even factors! 32 = 25, so 32 is “built from” five prime factors of 2 and nothing else! Integer m is therefore a member of an “evens only” club, and the correct answer is D.

## Official GMAT problem for practice

If x, y, and z are positive integers such that x is a factor of y, and x is a multiple of z, which of the following is NOT necessarily an integer?

(A) (x + z) / z

(B) (y + z) / x

(C) (x + y) / z

(D) xy / z

(E) yz / x

We should immediately apply the logic covered in this article and note that since z is a factor of a factor (x) of y, z is a factor of y. x is divisible by z, and y is divisible by both x and z. Let’s take the answer choices one at a time. We are looking for the one whose numerator is not necessarily divisible by its denominator.

(A) (x + z) / z

This isn’t the answer we’re looking for. x is a multiple of z. Another way to think of this is that x is “built from” some number of z’s. So the numerator (x + z) could be represented as nothing but a group of z’s, which is certainly divisible by the denominator of z.

(B) (y + z) / x

This is our answer. Since y is a multiple of z, the numerator can be expressed as a group of z’s. But we don’t know from any given facts whether this group will be divisible by x. Let’s check the other answers for good measure.

(C) (x + y) / z

Since both x and y can be represented as groups of z’s, this expression is an integer. B is still looking good.

(D) xy / z

z is a factor of both x and y, so of course the product xy is divisible by z.

(E) yz / x

y is already divisible by x and doesn’t even need z’s help. yz is divisible by x, and answer choice B is still the only expression that might not be an integer.

At this point in our series, you should be developing some fluency with factor/multiple relationships. In the next article, we will explore what happens when one integer is NOT divisible by another: remainders.

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Contributor: Elijah Mize (Apex GMAT Instructor)