Manhattan GMAT Challenge Problem of the Week – 19 Oct 2010

by Manhattan Prep, Oct 19, 2010

Here is a new Challenge Problem! If you want to win prizes, try entering our Challenge Problem Showdown. The more people enter our challenge, the better the prizes.

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

(A) 14,173

(B) 14,361

(C) 14,643

(D) 14,737

(E) 14,981

Answer

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.

Lets 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:

[pmath](2k + 1)^2[/pmath]

=[pmath]4k^2 + 4k + 1[/pmath]

Since the first two terms ([pmath]4k^2[/pmath] and [pmath]4k[/pmath]) 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 + [pmath]121^2[/pmath].)

The correct answer is (C).

Special Announcement: If you want to win prizes for answering our Challenge Problems, try entering our Challenge Problem Showdown. Each week, we draw a winner from all the correct answers. The winner receives a number of our our Strategy Guides. The more people enter, the better the prize. Provided the winner gives consent, we will post his or her name on our Facebook page.