Modular Arithmetic

Modular Arithmetic Created by previous Honors Pre-Calculus students

Division Algorithm • If a and b are integers where b > 0, There exist integers, q and r, with the property that: a = bq + r where 0 ≤ r < b. • For example, if a = 17 and b = 5, 17 = 5∙3 +2. Thus, q = 3 & r = 2. r=2 is called the remainder.

Modular Arithmetic • Modular arithmetic is an application of the division algorithm • For example, if it’s now May, what month will it be 25 months from now? • You got an answer of June right? • You probably didn’t count did you? You observed, 25 = 2∙12 + 1 • So you just added one month to the current month to get your answer of June.

Try a couple… • counting in hours - if it is 10:00 a.m. what time will it be 30 hours from now? • counting in days - if it is Wednesday what day of the week will it be 452 days from now? • counting degrees - if you are facing north and spin clockwise 810 degrees, which way are you facing?

Counting Answers • We are only interested in the fact that 30 is 6 more than 24 and that 10 (a.m.) plus 6 leaves a remainder of 4 when we take away 12. 4 pm • in the second case that 452 leaves a remainder of 4 when we take away multiples of 7 Sunday 3. 810 leaves a remainder of 90 when we take away multiples of 360. East

Clock Arithmetic = Modular Arithmetic • Click on the following link to see a visual demonstration of clock arithmetic or display of numbers mod 12. Check out the part about negative numbers! • http://www.math.csusb.edu/faculty/susan/number_bracelets/mod_arith.html

Modular Arithmetic Continued… • When a = qn + r, where q is the quotient and r is the remainder upon dividing a by n, we write: a mod n = r OR r = a modulo n • n is the modulus. Sometimes r is called the residue • For example: • 17 mod 5 = 2 because 17 = 5∙3 + 2 • 35 mod 7 = 0 because 35 = 7∙5 + 0 • 29 mod 8 = 5 because 29 = 8∙3 + 5

Notice… Notice that when you are modding by 12 your remainders (the answers) will be between 0 and 11. You will never get an answer of 12 because that means that you could have divided further. For example: 36 mod 12 = 0 (it goes in evenly)

Try a few on your own 1. 8 mod 13 2. 23 mod 11 3. 46 mod 7 4. 42 mod 3 5. 58 mod 4 6. 92 mod 15 7. 27 mod 11 8. 84 mod 5

Answers 1. 8 mod 13 = 8 because 8 = 0∙13 + 8 2. 23 mod 11 = 1 because 23 = 2∙11 + 1 3. 46 mod 7 = 4 because 46 = 6∙7 + 4 4. 42 mod 3 = 0 because 42 = 14∙3 + 0 5. 31 mod 8 = 7 because 31 = 3∙8 + 7 6. 92 mod 15 = 2 because 92 = 6∙15 + 2 7. 27 mod 11 = 5 because 27 = 2∙11 + 5 8. 84 mod 5 = 4 because 84 = 16∙5 + 4

And a few more… 1.) -5 mod 12 2.) -4 mod 10 3.) -15 mod 15 4.) -23 mod 8 5.) -28 mod 7 6.) -46 mod 4 7.) -50 mod 9 8.) -61 mod 3

And a few more… 1.) -5 mod 12 = 7 because -5 = 12∙ -1 + 7 2.) -4 mod 10 = 6 because -4 = 10∙ -1 + 6 3.) -15 mod 15 = 0 because -15 = 15∙ -1 + 0 4.) -23 mod 8 = 1 because -23 = 8∙ -3 + 1 5.) -28 mod 7 = 0 because -28 = 7∙ -4 + 0 6.) -46 mod 4 = 2 because -46 = 4∙ -12 + 2 7.) -50 mod 9 = 4 because -50 = 9∙ -6 + 4 8.) -61 mod 3 = 2 because -61 = 3∙ -21 + 2

Congruences • This idea of congruence was first developed by the mathematician Carl Friedrich Gauss in the late 18th century. • a ≡ b if a = b mod n • For example: 24 ≡ 9 mod 5 because 24 mod 5 = 4 and 9 mod 5 = 4

Good websites • http://mathcentral.uregina.ca/QQ/database/QQ.09.98/kupper1.html

Modular Arithmetic