Traditional books on number theory that emphasize the algebraic aspects begin with the subject of modular arithmetic. This is a very powerful technique, based on the work of Fermat, Euler, and Gauss. The idea is to fix an integer, n, and group all the rest of the integers into one of n classes, depending on what remainder you get when you divide by n. A more elegant way of saying this is that two integers, a and b, are in the same class if n divides b − a or, in other words, b − a is an integer multiple of n. In this case, we write a ≡ b mod n. (This is pronounced “a is congruent to b modulo n.”) For example, with n = 6, 5 ≡ 17 mod 6, −2 ≡ 4 mod 6, and even 6 ≡ 0 mod 6. Because there are seven days in a week, the fact that 3 ≡ 10 mod 7 means that the 3rd of the month and the 10th of the month fall on the same weekday.

Example 14.0.1. The following steps show that ≡ is an equivalence relation.

1. Show that ≡ is reflexive, that is, that a ≡ a mod n.

2. Show that ≡ is symmetric; that is, if a ≡ b mod n, then b ≡ a mod n.

3. Show that ≡ is transitive; that is, if a ≡ b mod n and b ≡ c mod n, then a ≡ c mod n.