Now here we are going to discuss a new type of addition, which is known as “addition modulo m” and written in the form , where and belong to an integer and is any fixed positive integer.
By definition we have
Here is the least non-negative remainder when , i.e., the ordinary addition of and is divided by .
For example, , since , i.e., it is the least non-negative reminder when is divisible by .
Thus to find , we add and in the ordinary way and then from the sum, we remove integral multiples of in such a way that the remainder is either or a positive integer less than .
When and are two integers such that is divisible by a fixed positive integer , then we have . This is read as is concurrent to .
Thus, if and only if is divisible by . For example since is divisible by , , , .