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∙  So you just added one month to the current month to get your answer of June.

Try a couple… 1. counting in hours - if it is 10:00 a.m. what time will it be 30 hours from now? 2. counting in days - if it is Wednesday what day of the week will it be 452 days from now? 3. counting degrees - if you are facing north and spin clockwise 810 degrees, which way are you facing?

Counting Answers 1. 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 pm 2. in the second case that 452 leaves a remainder of 4 when we take away multiples of 7 Sunday 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! mber_bracelets/mod_arith.html mber_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 = rOR 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 = 5because 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 mod mod mod mod mod mod mod 5

Answers 1.8 mod 13 = 8 because 8 = 0∙ mod 11 = 1 because 23 = 2∙ mod 7 = 4 because 46 = 6∙ mod 3 = 0 because 42 = 14∙ mod 8 = 7 because 31 = 3∙ mod 15 = 2 because 92 = 6∙ mod 11 = 5 because 27 = 2∙ 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= 7because -5 = 12∙ )-4 mod 10 = 6because -4 = 10∙ )-15 mod 15 = 0because -15 = 15∙ )-23 mod 8 = 1because -23 = 8∙ )-28 mod 7 = 0because -28 = 7∙ )-46 mod 4 = 2because -46 = 4∙ )-50 mod 9 = 4because -50 = 9∙ )-61 mod 3 = 2because -61 = 3∙

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 Q.09.98/kupper1.html Q.09.98/kupper1.html