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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