Is a positive integer even if its square is even?

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Let's take a positive integer n.

We can write it as a product of prime numbers:

n=p1p2...pr, where p1, p2, ..., pr are prime factors of n.

Now, assume that n2 is even. Then one of the pi equals 2. Why?

Note that n2=p12...pr2. Also, since n2 is even, then n2=2k for some k positive integer. 

=> 2k=p12...pr2, which, since 2 is a prime, implies that 2 = pi for some i.

Hence, n=2p1p2...pi-1pi+1...pr.

And so, n is even.

 

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