Yesterday, Manhattan GMAT posted a 700 level GMAT question on our blog. Today, they have followed up with the answer:
First, we should understand what the question is asking for. What is “the remainder, after division by 100” of a large integer? Test some numbers, if necessary:
321 divided by 100 leaves a remainder of 21.
432 divided by 100 leaves a remainder of 32.
Thus, we can see that the remainder, after division by 100, of a large integer is just the two-digit number formed by the last two digits of the integer (the tens digit and the units digit).
Now, we turn our attention to the actual integer in question, 710. This is a power of 7, so we need to look for any pattern in the last two digits of the powers of 7. (We can assume that there must be such a pattern; otherwise, this question would not be realistically solvable on the GMAT.)
71 = 7
72 = 49
73 = 49 × 7 = 343. Note that we only need to pay attention to the last 2 digits (43), so we will write …43.
74 = …43 × 7 = …01.
75 = …01 × 7 = …07.
At this point, we see that the cycle is starting to repeat. The next power (76) will end in …49, and so forth. Since we only have to go to 710, we may as well just keep going:
77 = …43
78 = …01
79 = …07
710 = …49
The correct answer is D.