Discrete Mathematical Structures by Pearson

Algebraic Structures 7-17

{4} =

4914 19

1611 16

,,

,,

,,

,,



−− −−













7.4.2 Addition and Multiplication of Residue Classes

Definition 7.4 Let {r

1

} and {r

2

} be two residue classes mod m, where 0 ≤ r

1

≤ m - 1 and 0 ≤ r

2

≤ m - 1.

The sum is written as

{r

1

} + {r

2

} = {r

1

+ r

2

}

where r

1

+ r

2

must be reduced to mod m if r

1

+ r

2

≥ m.

Example 1 Let us consider residue classes modulo 6, that is,

{0}, {1}, {2}, {3}, {4}, {5}.

Then,

{3} + {5} = {8} = {2}

{2} + {4} = {6} = {0}

{4} + {5} = {9} = {3}

Definition 7.5 The product of residue classes {r

1

} and {r

2

} mod m is defined as follows:

{r

1

}.{r

2

} = {r

1

r

2

}, 0 ≤ r

1

≤ m - 1, 0 ≤ r

2

≤ m - 1

where r

1

r

2

is reduced to mod m where r

1

r

2

≥ m.

THEOREM 7.13 The residue classes modulo m form a group with respect to addition ...