Today, let’s look at a GRE practice Quantitative Reasoning problem that deals with basic number properties such as odd/even and positive/negative, something that will help your GRE practice on the math section.
THE PROBLEM:
(x - y) (x - y) (x - y) (x - y) (y - x) = -28,629,151
Given that the above equation is true, and that x and y are integers, which of the following statements must also be true? Select all that apply.
A. (x - y) is divisible by 3
B. x is even
C. y is odd
D. (y - x) is odd
E. (x - y) < 0
THE SOLUTION
The answer choices seem to cover three basic concepts of number properties. Choices B, C, and D are all concerned with whether certain numbers are even or odd. Choice E deals with whether a quantity is negative (less than zero). And Choice A deals with divisibility.
Which area do we address first? Let’s look at what we know: On the left side of the equation are five factors multiplied by each other. Each factor contains the difference between x and y. On the right side of the equation is an 8-digit number. Very quickly we can note 2 important facts about the 8-digit number: it’s negative, and it’s odd.
Let’s deal with the negative/positive aspect of the problem first. Is the quantity (x - y) < 0? Or in other words, is (x - y) negative? Taking a quick look at the equation, it looks like (x - y) is repeated 5 times. If we were to restate that as (x - y) raised to the 5th power, we would be making a big mistake, however.
Look carefully at the last factor. You may have already noticed, but if it escaped your attention the first time, the fifth factor is different from the first four. Instead of being “(x - y)”, it is “(y - x).”
So the left hand side could be restated as “the quantity (x - y) raised to the fourth power, times the quantity (y - x) raised to the first power.” Focusing on the powers, we have an exponent of 4 and an exponent of 1. And we know that any quantity raised to an even-numbered power has to yield a positive number. Any quantity raised to an odd-numbered power preserves its original +/- sign.
The nice thing about thinking exponentially is that now we can picture -28,629,151 as being the product of just two factors, instead of five. That simplifies keeping track of what the sign should be: to get an negative answer when we multiply two numbers, the two numbers must have opposite signs. Since (x - y) raised to the fourth power must be positive, that means that (y - x) has to be a negative quantity. So (y - x) < 0. However, when we reverse the order of subtraction, (x - y) > 0. So Choice E is false and must be eliminated.
Now let’s look at the odd/even aspect of the problem.
When we multiply 5 factors together and get an odd number, that means that all of the factors are odd. This is an example of a set of rules that you’ve probably seen before. When multiplying odd and even numbers, the following patterns always apply:
even x even = even
even x odd = even
odd x odd = odd.
REMEMBER: ALL of the integer factors of an odd number MUST be odd.
So we know that the quantity (x - y) is odd, and the quantity (y - x) is also odd. That means we already know that Choice D is correct. What about choices B and C?
It may help here to compare (x - y) and (y - x) to the subtraction rules for odd/even numbers:
odd - even = odd
even - even = even
even - odd = odd
Notice that according to the rules, there are two different scenarios in which the difference between x and y is an odd number. According to the first rule, if x is odd and y is even, their difference will be odd. According to the third rule, if x is even and y is odd, their difference is still odd.
The second rule tells us that, regarding x and y, one of them has to be even, and one of them has to be odd; otherwise rule one or rule three would apply and the difference would be even. However, the question doesn't give us any extra information about x and y to tell us exactly which one is even and which one is odd. Since neither Choice B nor Choice C has to be true, we can eliminate both, since the question asks which answer choice/s MUST be true, rather than which answer choice/s may sometimes be true.
Now to evaluate whether Choice A is true. Is (x - y) divisible by 3? Or to restate it slightly, is 3 a factor of (x - y)? To help us solve this, we could start by plugging the big number into the calculator and see if it divides evenly by 3.
However, there’s a neat trick you can use to test for divisibility by 3 without using a calculator. Just add the separate digits of 28,629,151 like so:
2 + 8 + 6 + 2 + 9 + 1 + 5 + 1 = 34.
34 is not a multiple of 3, which means that 28,629,151 is not a multiple of 3 either. But are we finished yet? It’s good to be a bit cautious near the end and asked ourselves if we missed anything. The question is whether (x - y) is divisible by 3, not whether -28,629,151 is divisible by 3.
However, after thinking about it for a moment, we can feel confident that we really are ready to judge whether Choice A is true. If a factor of 3 is not found in 28,629,151, then it is impossible for a factor of 3 to be included in any of the five factors on the left side of the equation (remember that each “(x - y)” or “(y - x)” represents an actual numerical factor of -28,629,151). So we have to eliminate Choice A.
Only Choice D is a correct answer.
Good luck with your GRE practice!
