Now here we going to discuss with a new type of addition which is known as “addition modulo m” and written in the form where and belongs to an integer and is any fixed positive integer.
By definition we have
Where is the least non-negative remainder when , i.e., the ordinary addition of and , is divided by .
For example, , since , i.e., 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 reminder is either or a positive integer less than .
When and are two integer such that is divisible by a fixed positive integer, then we have . Which is read as is concurrent to .
Thus, if and only if is divisible by . For example since is divisible by , , , .