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Question

Prove that n2n is divisible by 2 for every positive integer.

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Solution

Any positive integer is of the form 2q or 2q+1, where q is some integer
When, n=2q,
n²+n=(2q)²+2q
=4q²+2q
=2q(2q+1)
which is divisible by 2$
when n=2q+1
n²n=(2q+1)²+(2q+1)
=4q²+4q+1+2q+1
=4q²+6q+2
=2(2q²+3q+1)
which is divisible by2
hence n²+n is divisible by 2 for every positive integer n.

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