If m ≡ n(mod4) and m ≡ n (mod6), does it follow that ≡ n (mod24)?

3 Congruence

Congruences are an important and useful tool for the study of divisibility. As we shall see, they are also critical in the art of cryptography.

Definition 3.1 If a and b are integers and n > 0, we write a ≡ b mod n

to mean n|(b − a). We read this as “a is congruent to b modulo (or mod) n. For example, 29 ≡ 8 mod 7, and 60 ≡ 0 mod 15.

The notation is used because the properties of congruence “≡” are very similar to the properties of equality “=”. The next few result make this clear.

Theorem 3.2 For any integers a and b, and positive integer n, we have: 1. a ≡ a mod n.

2. Ifa≡bmodnthenb≡amodn.

3. Ifa≡bmodnandb≡cmodnthena≡cmodn

These results are classically called: 1. Reflexivity; 2. Symmetry; and 3. Transitivity. The proof is as follows:

1. n|(a − a) since 0 is divisible by any integer. Therefore a ≡ a mod n.

2. If a ≡ b mod n then n|(b − a). Therefore, n|(−1)(b − a) or n|(a − b). Therefore, b ≡ a mod n.

3. If a ≡ b mod n and b ≡ c mod n then n|(b−a) and n|(c−b). Using the linear combination theorem, we have n|(b−a+c−b) or n|(c−a). Thus, a≡c mod n.

The following result gives an equivalent way of looking at congruence. It replaces the con-

Step-by-step explanation: