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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

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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

by Vincen » Sat Nov 27, 2021 4:03 am

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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
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Re: If the quantity \(5^2 + 5^4 + 5^6\) is written as \((a + b)(a - b),\) in which both \(a\) and \(b\) are integers, wh

by regor60 » Sat Nov 27, 2021 5:30 am
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
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