HP 48gII Graphing Calculator Uživatelský manuál stáhnout pdf (Strana 185)

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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 × 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.

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