Which of the following is two more than the square of an odd integer?

MANHATTAN GMAT OFFICIAL SOLUTION:

From the size of the answer choices, we should recognize that there must be some trick – a way to find the answer quickly within 2 minutes, since theoretically, every GMAT problem must be solvable that way.

Let’s start from first principles. We can represent a general odd integer as 2k + 1, where k is an integer. So the square of an odd integer can be written this way:

(2k + 1)^2

= 4k^2 + 4k + 1

Since the first two terms (4k^2 and 4k) contain 4 as a factor, we can see that they are both multiples of 4, and thus their sum is also a multiple of 4. This means that the square of any odd integer is 1 more than a multiple of 4. Thus, the number we are looking for (“two more than the square…”) is 3 more than a multiple of 4.

To quickly determine whether a number is a multiple of 4, we can just examine the last 2 digits. The reason is that 100 is divisible by 4, so only the tens and the ones digit matter. The hundreds digit, the thousands digit, etc. will never affect divisibility by 4.

Looking at the answer choices and cutting off everything but the last 2 digits, we are left with 73, 61, 43, 37, and 81. Of these, only 43 is 3 more than a multiple of 4 (40). Thus, C is the right answer. (Incidentally, 14,643 is 2 + 121^2.)

The correct answer is (C).