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
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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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
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