M7MBA wrote: ↑Sun Jun 14, 2020 1:52 pm
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
[spoiler]OA=E[/spoiler]
Source: Manhattan GMAT
So, we have \(5^2 + 5^4 + 5^6 = (a + b)(a – b),\)
\(5^2 + 5^4 + 5^6 = 5^2(1 + 5^2 + 5^4)\)
\(5^2(1 + 5^2 + 5^4) = 25(1 + 25 + 525)=25*651\)
So, we have \((a + b)(a – b) = a^2-b^2 = 25*651\)
Let's try option values.
A. 5 => \(b^2 = 25 => a^2-b^2 = a^2 - 25 = 25*651 => a^2 = 25*651 + 25 = 25*652 => a = 5*√652\). This cannot be the answer as it does not give a as an integer.
B. 10 => \(b^2 = 10 => a^2-b^2 = a^2 - 100 = 25*651 => a^2 = 25*651 + 100 = 25*655 => a = 5*√655\). This cannot be the answer as it does not give a as an integer.
C. 15 => \(b^2 = 225 => a^2-b^2 = a^2 - 225 = 25*651 => a^2 = 25*651 + 225 = 25*660 => a = 5*√660\). This cannot be the answer as it does not give a as an integer.
D. 20 => \(b^2 = 400 => a^2-b^2 = a^2 - 400 = 25*651 => a^2 = 25*651 + 400 = 25*667 => a = 5*√667\). This cannot be the answer as it does not give a as an integer.
E. 25 => \(b^2 = 625 => a^2-b^2 = a^2 - 625 = 25*651 => a^2 = 25*651 + 625 = 25*676 => a = 5*√676 = 5*26= 130\). This is the answer as gives a as an integer.
Correct answer:
E
Hope this helps!
-Jay
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