Question

Please help!
Prove: If n is congruent to 1(mod 6), then n^2 + 2^n is composite. Please help!
Prove: If n is congruent to 1(mod 6), then n^2 + 2^n is composite. @Mathematics

Answer 1

If we're assuming that n^2 + 2^n is composite, then ab = n^2 + 2^n (mod 6) where a and b are not equal to 1

So,

ab = n^2 + 2^n (mod 6)


ab = 1^2 + 2^1 (mod 6)


ab = 1 + 2 (mod 6)


ab = 3 (mod 6)


Now let a = 3 and b = 3, since ab = 3*3 = 9 = 3 (mod 6), this means that we've shown that 3 is composite mod 6


Therefore, n^2 + 2^n is composite mod 6 when n = 1 (mod 6)