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

This topic has expert replies
Legendary Member
Posts: 2898
Joined: 07 Sep 2017
Thanked: 6 times
Followed by:5 members

Timer

00:00

Your Answer

A

B

C

D

E

Global Stats

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

Answer: E

Source: Manhattan GMAT

Master | Next Rank: 500 Posts
Posts: 409
Joined: 15 Oct 2009
Thanked: 27 times
Factor out 5^2:

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

The expression in parentheses is close to the form (a+b)(a+b) = a^2+2ab+b^2
where a=5^2 and b=1, except it yields one more 5^2 than in the parentheses.

Let's adjust for that by subtracting out the extra 5^2:
5^2(5^2+1)^2 - (5^2)^2

This is a difference of squares:

[5(5^2+1)+5^2]* [5(5^2+1)-5^2]

Don't need to solve this to see that b=25,E