GRE Math for People Who Hate Math: Which of the Following is a Factor of x?
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Did you know that you can solve ‘which of the following is a factor’ problems with hardly any math at all? It just takes a little basic arithmetic, logical reasoning, and creative thinking — skills that you already have.
Take a quick look at this problem:
n is divisible by 14 and 21. Which of the following statements must be true? Indicate all such statements.
[] 2 is a factor of n.
[] 6 is a factor of n.
[] 8 is a factor of n.
[] 49 is a factor of n.
For a moment, ignore the math in this problem entirely. Instead, imagine that two people are at a picnic, and they each take turns opening the cooler to see what drinks are available.
Amanda saw a bottle of root beer and a bottle of cola in the cooler. Jordan saw a bottle of cola and a bottle of lemonade in the cooler. Assuming that Amanda and Jordan both told the truth, which of the following statements must be true? Indicate all such statements.
[] A bottle of root beer is in the cooler.
[] A bottle of root beer and a bottle of lemonade are in the cooler.
[] Three bottles of root beer are in the cooler.
[] Two bottles of cola are in the cooler.
Logically, the first two statements are true. But the next two statements aren’t necessarily true. Nobody saw three bottles of root beer, so you don’t know that they’re there. And, even though Amanda and Jordan both saw a bottle of cola, they might have actually seen the same bottle. There might only be one bottle of cola, not two.
That problem is exactly analogous to the math problem at the beginning of this article. Think of the mystery number, n, as the cooler. The contents of the cooler are the prime factors of n. Since 14 divides into n, someone saw its prime factors, 2 and 7, in the cooler. Someone else saw the prime factors of 21, 3 and 7. There must be a 2, a 3, and a 7 in there. But, do you know that there are two 7s, making 49? No, because you could have just seen the same 7 twice.
Amanda opened the cooler, dug through the ice a bit, and said, “I see three bottles of root beer and one bottle of lemonade.”
Then, Jordan opened the cooler, and said, “I can’t find four bottles of root beer in here.”
Assuming Amanda and Jordan both told the truth, which of the following cannot be in the cooler?
(A) three bottles of root beer
(B) three bottles of lemonade
(C) five bottles of root beer
(D) two bottles of lemonade and two bottles of root beer
(E) three bottles of lemonade
If you picked (C), you’re correct. Jordan told us that he couldn’t find four bottles of root beer. So, there certainly can’t be five bottles of root beer.
Once again, think of prime factors as bottles of beverages. Here’s the same problem, written out in GRE terms:
k is a multiple of 24 but not a multiple of 16. Which of the following cannot be a factor of k?
(A) 8
(B) 9
(C) 32
(D) 36
(E) 81
If k is a multiple of 24, that means you saw all of the prime factors of 24 in the cooler. The cooler contains 2, 2, 2, and 3: three bottles of root beer, and one bottle of lemonade. But, because k isn’t a multiple of 16, then you know that the cooler doesn’t contain the prime factors of 16. It doesn’t contain 2, 2, 2, and 2: four bottles of root beer.
What could the cooler contain, given what you know? It definitely couldn’t contain five 2s. (C) is right, since 32 contains five 2s, but k doesn’t.
Think of the prime factors of a number as bottles in a cooler. Each time someone opens the cooler and tells you what they see, you learn something new about its contents. Unfortunately, you can’t just add what each person tells you together to figure out the contents of the cooler. Instead, you have to apply logic to determine what must be in there.
Take a look at the Divisibility & Primes chapter of the 5lb. Book of GRE Practice Problems, and see if you can come up with some other problems to use this analogy on. It’s useful for a wide range of problems, and as you practice using it, you’ll find that the math will come more and more naturally. In the meanwhile, feel free to share your ideas in the comments! 📝
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Chelsey Cooley
is a Manhattan Prep instructor based in Seattle, Washington. Chelsey always followed her heart when it came to her education. Luckily, her heart led her straight to the perfect background for GMAT and GRE teaching: she has undergraduate degrees in mathematics and history, a master’s degree in linguistics, a 790 on the GMAT, and a perfect 170/170 on the GRE. Check out Chelsey’s upcoming GRE prep offerings here.