prove for every positive integer n that n4 is odd if and onl

prove for every positive integer n, that n^4 is odd if and only if n is odd

n is odd

n = 2k+1 for some integer k

n^2 = 2(2k^2 +2k) +1

n^2 is odd

i.e

n is odd => n^2 is odd

n^2 is odd => n^4 is odd

i.e

n is odd => n^4 is odd..............(1)

let n^4 is odd .

if n is even

n =2k for some integer k

n^4 = 16k^4 = 2(8k^4) => n^4 is even which is a contradiction to our assumption n is even

n is odd

i.e

n^4 is odd => n is odd............(2)

(1) ,(2)combine to prove

n^4 is odd <=>n is odd

thus proved