In this chapter we are going to see some further results about congruence ofintegers. Most of these are to do with working out the congruence of largepowers of an integer modulo some given integer m. I showed you some waysof tackling this kind of question in the last chapter (see Examples 13.2 and13.3). The first result of this chapter — Fermat’s Little Theorem — is a generalfact that makes powers rather easy to calculate when m is a prime number. Therest of the chapter consists mainly of applications of this theorem to solvingsome special types of congruence equations modulo a prime or a product oftwo primes, and also to the problem of finding large prime numbers using acomputer. We’ll make heavy use of all this material in the next chapter onsecret codes.