Congruence Module

Question

Find all the values x (mod 13) satisfying x^12 + 2x + 1 ≡ 0 (mod 13)

Answer 1

Since 13 is prime then for any x≠0 mod 13 we have x^13-1 = x¹² = 1

so the equation x¹² +2x+1 = 0 (mod 13)

reduces to 1+2x+1=2x+2 = 0 (mod 13).

You should get the one solution.

Or you can try each number (1 to 12) relatively prime to 13.