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Question

For all positive integral values of n, 32n−2n+1 is divisible by

A
2
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B
4
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C
8
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D
12
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Solution

The correct option is A 2
32n=(1+2)2n=2nC0+2nC12+2nC222+........+2nC2n22n
=1+4n+2nC222+........+2nC2n22n
32n2n+1=2+2n+2nC222+........+2nC2n22n
=2(1+n+2nC22+........+2nC2n22n1)=2× some integer
Hence for all natural number 32n2n+1 is always divisible by 2

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