Cryptology - Unofficial St. Mary's College of California Web Site

3.1. THE REMAINDER OPERATOR 37
As a bit of a time saver, notice that after doing the division you do not need
to clear the screen (on your calculator) before punching in the decimal, but
can instead subtract off the quotient. For example, redoing 888%31, we have
888 ÷ 31 = 28.6451. Subtract 28 to leave .6451 which we then multiply by 31. 3
Examples: Compute the following remainders.
(1) 424%26 =
424/26 = 16.3077. Subtracting 16 leaves .3077. Answer is .3077 × 26 = 8.
(2) 58%7 =
58/7 = 8.2857 → .2857 × 7 = 2. Answer: 2.
(3) 101%11 =
(4) 2045%21 =
(5) 48924%30 = 4
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What about −29%12 Following our procedure (divide by 12, discard the
quotient, multiply by 12) gives −5. Since the goal was a small positive number,
we now simply add 12. So −29%12 = 7. In the shortcut notation, −29/12 =
−2.4167 → −2.4167 + 2 = −.4167 → −.4167 × 12 = −5 → −5 + 12 = 7.
Examples: Compute the following remainders.
(1) −89%23 =
−89/23 = −3.869 → −3.869 + 3 = −.869 → −.869 × 23 = −20 →
−20 + 23 = 3. Ans: 3
(2) −134%17 =
(3) −748%73 = 5
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Let’s summarize our work.
3 The presence of division, and the comment about rounding, might make one concerned
about the validity of this algorithm. Indeed, calculators only hold so many digits, so error is
possible, especially when a is very large. These are valid concerns, but ones we will not worry
about.
4 (4) 12, (5) 2, (6) 8, (7) 24
5 (2) 2, (3) 55