HP 50g User Manual page 190

multiplying j times k in modulus n arithmetic is, in essence, the integer
remainder of j k/n in infinite arithmetic, if j k>n. For example, in modulus 12
arithmetic we have 7 3 = 21 = 12 + 9, (or, 7 3/12 = 21/12 = 1 + 9/12, i.e.,
the integer reminder of 21/12 is 9). We can now write 7 3
read the latter result as "seven times three is congruent to nine, modulus twelve."
The operation of division can be defined in terms of multiplication as follows, r/
k j (mod n), if, j k
j k/n. For example, 9/7
divisions are not permitted in modular arithmetic. For example, in modulus 12
arithmetic you cannot define 5/6 (mod 12) because the multiplication table of
6 does not show the result 5 in modulus 12 arithmetic. This multiplication table
is shown below:
6*0 (mod 12)
6*1 (mod 12)
6*2 (mod 12)
6*3 (mod 12)
6*4 (mod 12)
6*5 (mod 12)
Formal definition of a finite arithmetic ring
The expression a
and holds if (b-a) is a multiple of n.
simplify to the following:
If
then
For division, follow the rules presented earlier. For example, 17
and 21 3 (mod 6). Using these rules, we can write:
17 + 21 5 + 3 (mod 6) => 38
17 – 21 5 - 3 (mod 6) =>
17 21 5 3 (mod 6) => 357
r (mod n). This means that r must be the remainder of
3 (mod 12), because 7 3
0
6
0
6
0
6
b (mod n) is interpreted as "a is congruent to b, modulo n,"
a b (mod n) and c
a+c
a-c b - d (mod n),
a c
-4
6*6 (mod 12)
6*7 (mod 12)
6*8 (mod 12)
6*9 (mod 12)
6*10 (mod 12)
6*11 (mod 12)
With this definition the rules of arithmetic
d (mod n),
b+d (mod n),
b d (mod n).
8 (mod 6) => 38
2 (mod 6)
15 (mod 6) => 357
9 (mod 12), and
9 (mod 12). Some
0
6
0
6
0
6
5 (mod 6),
2 (mod 6)
3 (mod 6)
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