When Logic Trumps Formulas: Even/Odd Integers
One such category of problems involve even/odd integer arithmetic. Although they involve few calculations, they can require a surprising amount of logical reasoning. Take this problem, for instance:
If a and b are integers, and m is an even integer, is ab/4 an integer?
(1) a + b is even.
(2) m/(ab) is an odd integer.
Before reading our solution, go ahead and attempt to solve this problem on your own.
The key to tackling this problem lies in using a few basic facts that you probably already know about even/odd integers. Here is a quick refresher on the rules:
- An even number can only be formed by the sum of either 2 odd numbers (odd + odd = even), or 2 even numbers (even + even = even).
- An odd number can only be formed by the sum of an odd and even number (odd + even = odd, or even + odd = odd).
- An even number can only be formed by multiplication in three ways: even·odd, odd·even, and even·even.
- An odd number can only be formed by multiplication in one way: odd·odd = odd.
The challenge is to use these four basic statements to deduce some hidden information.
Assuming that statement 1 alone is true, we know that a + b must be even, so according to rule #1, a and b must either both be even, or both be odd. Let’s consider the two possible scenarios:
Suppose a and b were both odd. We could plug in a = 3 and b = 1 to get ab/4 = 3·1/4 = 3/4; this is clearly not an integer. However, if both a and b were even, we could plug in a = 24 and b = 2, so that ab/4 = 24·2 / 4 = 12, which is in fact an integer. Because different plug-ins lead to different answers, we cannot answer with a definite yes or a definite no. Statement 1 alone is insufficient, so eliminate choices A and D.
Looking at statement 2 alone, we can assume that m/ab = odd. If we multiply both sides of by ab, we get m = ab·odd. Now, we’re told that m is an even integer, which means that even = ab·odd. According to rule #3, it is impossible to get an even number by multiplying two odd numbers; at least one number must be even. Therefore, since m is even, we must conclude that ab is also even.
If ab is even, does that give us any clue as to whether ab/4 must be an integer? If we try plugging in one set of numbers, such as m = 24, a = 4, and b = 2, then ab = 8, and ab/4 = 8/4 = 2. The answer for this set of plug ins is Yes, and so we might be tempted to conclude that the answer will always be Yes.
There is no such general rule, however, that states that an even number (ab) divided by another even number (4) will always be an integer. As an even integer, ab will be divisible by 2 without a remainder, but there is nothing that implies that it must be divisible by 4. For example, suppose a = 2, b = 1, and m = 6. In this case, ab = 2, and ab/4 = 2 / 4 = 1/2, which would result in a No answer to the question stem. Sometimes statement 2 gives us Yes, and other times No, so statement 2 alone is insufficient; we can proceed to eliminate choice B.
This changes when we look at both statements combined. We know from statement 1 that both a and b must either both be even or both odd, but statement 2 also tells us that ab is even. The only way to reconcile both statements is if both a and b are even; and if both are even, then they both must contain factors of 2. Their product, ab, will therefore contain two factors of 2, and so ab will contain the factor 2·2 = 4, and hence will be divisible by 4. This means that ab/4 must definitely be an integer. Since we have a definite Yes answer, both statements, when combined, are sufficient.
As you can see, problems with no formulas can still prove to be quite tricky. Often with the GMAT, it’s not about what formulas you remember, but about what you can logically deduce.
With that in mind, try out this challenge problem:
If a, b, and c are all integers, is odd?
(1) a + 2b + c is even.
(2) a·b·c is odd.
(Check back in a few days for the solution in the comments section!)