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If m and n are positive integers, is 36+36+m+n divisible by 4? 1) m

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If m and n are positive integers, is 36+36+m+n divisible by 4? 1) m [#permalink]

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If m and n are positive integers, is \(3^6+3^{6+m+n}\) divisible by 4?

1) \(m=3n+1\)
2) \(m+3n\) is an odd number
[Reveal] Spoiler: OA

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If m and n are positive integers, is 36+36+m+n divisible by 4? 1) m [#permalink]

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ziyuen wrote:
If m and n are positive integers, is \(3^6+3^{6+m+n}\) divisible by 4?

1) \(m=3n+1\)
2) \(m+3n\) is an odd number


\(3^6+3^{6+m+n}\) = \(3^6(1+3^{m+n})\)

the question is \(3^6+3^{6+m+n}\) divisible by 4 reduces to is \(3^6(1+3^{m+n})\) divisible by 4

since \(3^6\) is not divisible by 4 we have to find if \((1+3^{m+n})\) is divisible by 4

St I
m=3n+1

\((1+3^{m+n})\) = \((1+3^{3n+1+n})\) = \((1+3^{4n+1})\)

so for all the positive integer values of n, 4n+1 is an odd number and \(1+3^{Odd-Number}\) is always divisible by 4 -----------Sufficient

St II
m+3n is an Odd number

which means m+n is also Odd number and \(1+3^{Odd-Number}\) is always divisible by 4 ----------Sufficient

Hence option D is correct
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Re: If m and n are positive integers, is 36+36+m+n divisible by 4? 1) m [#permalink]

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New post 24 Aug 2017, 10:20
\(3^{6}+3^{6+m+n}\) = \(3^{6}\)(1+\(3^{m+n}\))
Thus if m+n is whether even or odd can give us a unique answer.
Statement 1:
m=3n+1
m-3n = 1 (odd), only possible when one of either m or n is odd and other is even. Which implies - m+n = odd.
Sufficient
Statement 2:
m+3n is odd,only possible when one of either m or n is odd and other is even. Which implies - m+n = odd.
Sufficient.
Thus option D
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Re: If m and n are positive integers, is 36+36+m+n divisible by 4? 1) m   [#permalink] 24 Aug 2017, 10:20
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If m and n are positive integers, is 36+36+m+n divisible by 4? 1) m

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