By unique factorization of integers, there is a smallest multiple of such that is exactly divisible by . Now apply the above argument with and replaced by and .

For example, are close in the -adic metric if their difference is divisible by a large power of . E.g., if then and are close because their difference is , which is divisible by a large power of .

We can finally define the -adic numbers.

William Stein 2012-09-24