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 $5$, $5 \equiv 5\left( {\bmod 5} \right)$, $16 \equiv 4\left( {\bmod 6} \right)$, $- 20 \equiv 4\left( {\bmod 6} \right)$.