Question

Show that n2+n+1 is not divisible by s for any n where n is a natural number

Answer 1

let n2=q and then New number =q+n+1 and by Euclid lemma here remainder is n+1 so it is not divided

Answer 2

5∤(n2+n+1)  for all integer n.

Because  n2+n+1≡(n+3)2+2(mod5)

So if 5∣(n2+n+1) for some n, then n satisfies (n+3)2≡3(mod5). But 3 is not quadratic residue modulo 5. (You can check that 3 is not quadratic residue easily.)

If you don't know modular arithmetic you can also show it by induction.

Prove that if n2+n+1 is not divisible by 5, then also (n+5)2+(n+5)+1 is not divisible by 5.

Prove explicitly the base cases: n=1,n=2,n=3,n=4,n=5.

By induction this then holds for all n.