Even the purest of pure mathematics can have a crucial influence on practical problems. In this article we show how a topic in pure mathematics (modular arithmetic) originally pursued for its own interest only, turns out to have unexpected application to an area of communication theory (cryptography). The fact that at the present time it is easy to construct large prime numbers but very difficult to factorise large composite numbers has made it possible to devise simple codes which are uncrackable by known methods.