Is a positive integer even if its square is even?

Let's take a positive integer n.We can write it as a product of prime numbers:n=p₁p₂...p_r, where p₁, p₂, ..., p_r are prime factors of n.Now, assume that n² is even. Then one of the p_i equals 2. Why?Note that n²=p₁²...p_r². Also, since n² is even, then n²=2k for some k positive integer. => 2k=p₁²...p_r², which, since 2 is a prime, implies that 2 = p_i for some i.Hence, n=2p₁p₂...p_i-1p_i+1...p_r.And so, n is even.