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Prove that if n n is even then n 2 ≡ 0 n^2 \equiv 0 (mod 4 4 ).

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Prove that if nn is even then n2≡0n^2 \equiv 0 (mod 44).

proof

Using the EI rule, there is an integer kk such that n=2kn=2k.

QED

n2≡0n^2 \equiv 0 (mod 44)

n2=(2k)2=4k2n^2 = (2k)^2 = 4k^2

Using the direct proof technique, we assume nn is even.

The statement we need to prove is a universally quantified statement, so we consider an arbitrary integer nn.

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