Show that 4*(n^2 + 1) is never divisible by 11.?

Question

(abstract algebra)

Answer 1

It is sufficient to prove that n^2+1 is never divisible by 11 since 4 does not contribute anything to its divisibility by 11.

You use modulo 11 to test n^2.

0^2 = 0 mod 11
1^2 = 1 mod 11
2^2 = 4 mod 11
3^2 = 9 mod 11
4^2 = 5 mod 11
5^2 = 3 mod 11
6^2 = 3 mod 11
7^2 = 5 mod 11
8^2 = 9 mod 11
9^2 = 4 mod 11
10^2 = 1 mod 11
anything greater than 10 repeats itself from 0^2 = 0 mod 11.

So a perfect square n^2 is either 0,1,3,4,5,9 mod 11 for all n.
Therefore n^2+1 = 1,2,4,5,6,10 mod 11. None of them are 0 mod 11. Thus it will never be divisible by 11.

Answer 2

uhm, very few even numbers are divisible by eleven, unless one of the quotients or "n" represents 11...