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6 does not show the result 5 in modulus 12 arithmetic. This multiplication
table is shown below:
6*0 (mod 12) 0
6*6 (mod 12)
0
6*1 (mod 12) 6
6*7 (mod 12)
6
6*2 (mod 12) 0
6*8 (mod 12)
0
6*3 (mod 12) 6
6*9 (mod 12)
6
6*4 (mod 12) 0
6*10 (mod 12) 0
6*5 (mod 12) 6
6*11 (mod 12) 6
Formal definition of a finite arithmetic ring
The expression
a
≡
b (mod n)
is interpreted as “
a
is congruent to
b
, modulo
n
,” and holds if
(b-a)
is a multiple of
n
. With this definition the rules of
arithmetic simplify to the following:
If
a
≡
b (mod n)
and
c
≡
d (mod n),
then
a+c
≡
b+d (mod n),
a-c
≡
b - d (mod n),
a
×
c
≡
b
×
d (mod n).
For division, follow the rules presented earlier. For example, 17
≡
5 (mod 6),
and 21
≡
3 (mod 6). Using these rules, we can write:
17 + 21
≡
5 + 3 (mod 6) => 38
≡
8 (mod 6) => 38
≡
2 (mod 6)
17 – 21
≡
5 - 3 (mod 6) => -4
≡
2 (mod 6)
17
×
21
≡
5
×
3 (mod 6) => 357
≡
15 (mod 6) => 357
≡
3 (mod 6)
Notice that, whenever a result in the right-hand side of the “congruence”
symbol produces a result that is larger than the modulo (in this case,
n
= 6),
you can always subtract a multiple of the modulo from that result and simplify
it to a number smaller than the modulo. Thus, the results in the first case
8
(mod 6)
simplifies to
2 (mod 6
), and the result of the third case,
15 (mod 6)
simplifies to
3 (mod 6)
. Confusing? Well, not if you let the calculator handle
those operations. Thus, read the following section to understand how finite
arithmetic rings are operated upon in your calculator.
