Addition modulo

Now here we are going to discuss a new type of addition, which is known as “addition modulo m” and written in the form a{ + _m}b, where a and b belong to an integer and m is any fixed positive integer.

By definition we have

a{ + _m}b = r,\,\,for\,0 \leqslant r < m

Here r is the least non-negative remainder when a + b, i.e., the ordinary addition of a and b is divided by m.

For example, 5{ + _6}3 = 2, since 5 + 3 = 8 = 1\left( 6 \right) + 2, i.e., it is the least non-negative reminder when 5 + 3 is divisible by 6.

Thus to find a{ + _m}b, we add a and b in the ordinary way and then from the sum, we remove integral multiples of m in such a way that the remainder r is either 0 or a positive integer less than m.

When a and b are two integers such that a - b is divisible by a fixed positive integer m, then we have a \equiv b\left( {\bmod m} \right). This is read as a is concurrent to b (mod m).

Thus,a \equiv b\left( {\bmod m} \right) if and only if a - b is divisible by m. For example 13 \equiv 3\left( {\bmod 5} \right) since 13 - 3 = 10 is divisible by 5, 5 \equiv 5\left( {\bmod 5} \right), 16 \equiv 4\left( {\bmod 6} \right),  - 20 \equiv 4\left( {\bmod 6} \right).