Bunuel wrote:

If the quantity 5^2+ 5^4 + 5^6 is written as (a + b)(a – b), in which both a and b are integers, which of the following could be the value of b?

A. 5

B. 10

C. 15

D. 20

E. 25

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:One first natural move is to factor \(5^2\) out of the numeric expression:

\(5^2 + 5^4 + 5^6 = 5^2(1 + 5^2 + 5^4)\)

Meanwhile, \((a + b)(a – b)\) becomes \(a^2 – b^2\).

So, somehow we have to turn \(5^2(1 + 5^2 + 5^4)\) 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: \(5^4 + 5^2 + 1\).

This should look almost like a special product: \((x + y)^2 = x^2 + 2xy + y^2\), if \(x = 5^2\) and \(y = 1\). In fact, write that out:

\((5^2+1)^2 = 5^4 + 2(5^2) + 1\)

The expression we actually have, \(5^4 + 5^2 + 1\), is very close. Add \(5^2\) and subtract it as well:

\(5^4 + 5^2 + 1\)

\(= 5^4 + 5^2 + 1 + 5^2 – 5^2\)

\(= (5^4 + 5^2 + 1 + 5^2) – 5^2\)

\(= (5^4 + 2(5^2) + 1) – 5^2\)

\(= (5^2 + 1)^2 – 5^2\)

\(= 26^2 – 5^2\)

Almost there. Remember, we had 5^2 as well:

\(5^2 + 5^4 + 5^6 = 5^2(1 + 5^2 + 5^4) = 5^2(26^2 – 5^2)\)

\(= (26*5)^2 – (5*5)^2\)

\(= 130^2 – 25^2\).

Oher 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.

\(5^2 + 5^4 + 5^6 = a^2 – b^2\)

\(5^2 + 5^4 + 5^6 + b^2 = a^2\)

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):

\(5^2+ 5^4 + 5^6 + 25^2\)

\(= 5^2 + 5^4 + 5^6 + 5^4\)

\(= 5^2 + 2×5^4 + 5^6\)

\(= (5 + 5^3)^2\)

This is the only answer that fits.

The correct answer is E.
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