Hello could anyone help me prove this Let fgk and m be integ

Hello, could anyone help me prove this:

Let f,g,k and m be integers with m > 1. If gcd(k,m) = 1 and kf kg (mod m), then f g (mod m).

Aside from stating the assumtions I don\'t have a clue what to do next.

Solution

Recall that Fermat’s “little theorem” says that if p is prime and a is not a multiple of p, then a p1 1 mod p. This theorem gives a possible way to detect primes, or more exactly, non-primes: if for a certain a coprime to n, a n1 is not congruent to 1 mod n, then, by the theorem, n is not prime. A lot of composite numbers can indeed be detected by this test, but there are some that evade it. We give ourselves some notation and terminology to discuss them. For a fixed a > 1, we write F(a) for the set of positive integers n satisfying a n1 1 mod n. By Fermat’s theorem, F(a) includes all primes that are not divisors of a. If n F(a), then gcd(a, n) = 1, since, clearly, gcd(a n1 , n) = 1. Also, a n a mod n; the reverse implication is true provided that a and n are coprime. A composite number n belonging to F(a) is called an a-pseudoprime, or a pseudoprime to the base a. (Some writers require that n should also be odd, but we will not adopt this convention here.) 2-pseudoprimes are sometimes just called “pseudoprimes”. A number n that is a-pseudoprime for all a coprime to n is called a Carmichael number, in honour of R.D. Carmichael, who demonstrated their existence in 1912.