Addition and multiplication modulo m, a positive integer
(Monday, May 3, 2010)
We now define two types of composition known as addition modulo m and multiplication modulo m, where m is a fixed positive integer.
(a) If a, b are two integers and r is the
least non-negative remainder
obtained by dividing the sum a + b
by m, then we define the
composition, addition modulo m by the relation
a + b (mod m) = r, where 0 £ r £ m, for a, b e Z.
For example 13 + 7 (mod 6) = 2, since
13 +7 = 6 x 3 +2.
(b) If r be the least non-negative
remainder when the product a x b is
divided by m, then we define the
composition multiplication modulo
m by the relation
a x b (mod m) = r, where 0 £ r £ m,
for a, b e Z.
For example, 8 x 7 (mod 5) = 1, as
8x7=5x11+1
If s = {0,1, 2, 3, 4 , 5} then under addition modulo 6 and multiplication modulo 6 the composition tables are as follows.
+6 | 0 | 1 | 2 | 3 | 4 | 5 |
0 | 0 | 1 | 2 | 3 | 4 | 5 |
1 | 1 | 2 | 3 | 4 | 5 | 0 |
2 | 2 | 3 | 4 | 5 | 0 | 1 |
3 | 3 | 4 | 5 | 0 | 1 | 2 |
4 | 4 | 5 | 0 | 1 | 2 | 3 |
5 | 5 | 0 | 1 | 2 | 3 | 4 |
x6 | 0 | 1 | 2 | 3 | 4 | 5 |
0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 2 | 3 | 4 | 5 |
2 | 0 | 2 | 4 | 0 | 2 | 4 |
3 | 0 | 3 | 0 | 3 | 0 | 3 |
4 | 0 | 4 | 2 | 0 | 4 | 2 |
5 | 0 | 5 | 4 | 1 | 2 | 1 |
Posted in Posted by waytofeed at 7:40 AM
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