Prove that for all integers n n2 n 3 is oddSolutionn2 n3

Prove that for all integers n, n^2 + n + 3 is odd.

Solution

n² +n+3 = n²+2n+1 -n+2 = (n+1)² -(n-2)

N is even, n+1 is odd, n-2 is even, hence the first figure is square of a odd which is odd. and a even number (n-2) is subtracted. Since, difference between odd and even is a odd number, Hence,  (n)² +(n+3) is odd, when n is even

N is odd, n+1 is even, n-2 is odd, hence the first figure is square of a even which is even. and a odd number (n-2) is subtracted. Difference between a even and odd is odd number. Hence,  (n)² +(n+3) is odd, when n is odd

Hence, n² +n+3 is always odd