If n is an integer prove (n+3)^(2)-n^(2) is never even.
Let us begin by simplifying the expression:(n+3)2 - n2 = (n+3)(n+3) - n2= n2 + 6n + 9 - n2 (expanded brackets)= 6n + 9 (collected like terms)= 3(2n+3) (taken out a factor of 3)Now we can consider this simpler equivalent expression.3 is an odd number2n is even thus 2n+3 is odd (even plus odd is always odd)so we have an odd*odd which is always odd, thus never even and we are done.
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