HP 48gII - Manual (Page 185)

Page 5-14
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 Ч 21 ≡ 5 Ч 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.