Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
One first natural move is to factor 52 out of the numeric expression:
52 + 54 + 56 = 52(1 + 52 + 54)
Meanwhile, (a + b)(a – b) becomes a2 – b2.
So, somehow we have to turn 52(1 + 52 + 54) into a difference of squares. It must come down to the stuff in the parentheses.
Flip around those terms, so that we have the larger powers first: 54 + 52 + 1.
This should look almost like a special product: (x + y)2 = x2 + 2xy + y2, if x = 52 and y = 1. In fact, write that out:
(52 + 1)2 = 54 + 2(52) + 1
The expression we actually have, 54 + 52 + 1, is very close. Add 52 and subtract it as well:
54 + 52 + 1
= 54 + 52 + 1 + 52 – 52
= (54 + 52 + 1 + 52) – 52
= (54 + 2(52) + 1) – 52
= (52 + 1)2 – 52
= 262 – 52
Almost there. Remember, we had 52 as well:
52 + 54 + 56 = 52(1 + 52 + 54) = 52(262 – 52)
= (26×5)2 – (5×5)2
= 1302 – 252.
Other differences of squares can be equal to the original expression, but this is the only one that fits an answer choice: (E) 25.
There are other ways to solve this problem as well, such as backsolving.
52 + 54 + 56> = a2 – b2
52 + 54 + 56 + b2>= a2
Try different b’s from the answer choices, and see which one, when added to the original expression, gives you a perfect square.
Testing (E):
52 + 54 + 56 + 252
= 52 + 54 + 56 + 54
= 52 + 2×54 + 56
= (5 + 53)2
This is the only answer that fits.
The correct answer is E.