Question

For how many integers n > 1 is x^49 congruent to x (modulo n) true for all integers x?

Answer 1

Hint:
Firstly, note that, for relatively prime integers m and n, x^49 congruent to x (modulo mn) if and only if x^49 congruent to x (modulo m) and x^49 congruent to x (modulo n).

Next, by considering x^49 congruent to x (modulo p^r), where p is prime, r greater than or equal to 1, and x = p^r−1, show that r > 1 is impossible.