Addition & Multiplication of Modulo - Abstract Algebra

Now we are going to discuss a new type of addition which is known as “addition modulo m” and written in the form

where a and b belongs to an integer and m is any fixed positive integer.
By definition we have

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

, since

, i.e., is the least non-negative reminder when

is divisible by 6.
Thus to find

, 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 reminder

is either

or a positive integer less than m.
When

and

are two integer such that a-b is divisible by a fixed positive integer m, then we have

. Which is read as “a is concurrent to b mod m”.
Thus,

if and only if a-b is divisible by m. For example

since 13-10=10 is divisible by 5,

Now we are going to define a new type of multiplication which is known as “multiplication modulo p” and it can be written as

, where a and b are any integers and p is a fixed positive integer.
By Definition, we have

Where r is the least non-negative remainder when ab, i.e. the ordinary product of a and b, is divided by p. For example

, since

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