modular arithmetic
- 2. OBJECTIVES •UNDERSTAND MODULAR ARITHMETIC WITH EXAMPLES •UNDERSTAND ABOUT CONGRUENCE •IDENTIFY VALID AND INVALID CONGRUENCE •USE MODULO CONCEPTS AND PROPERTIES TO SOLVE MODULAR ARITHMETIC PROBLEMS AND COMPOSE PROOFS
- 3. MODULAR ARITHMETIC •ANOTHER APPLICATION OF THE DIVISION ALGORITHM THAT WILL BE IMPORTANT TO US IS MODULAR ARITHMETIC. MODULAR ARITHMETIC IS AN ABSTRACTION OF A METHOD OF COUNTING THAT YOU OFTEN USE
- 4. FOR EXAMPLE • IF IT IS NOW SEPTEMBER, WHAT MONTH WILL IT BE 25 MONTHS FROM NOW? • OF COURSE, THE ANSWER IS OCTOBER, BUT THE INTERESTING FACT IS THAT YOU DIDN’T ARRIVE AT THE ANSWER BY STARTING WITH SEPTEMBER AND COUNTING OFF 25 MONTHS. • INSTEAD, WITHOUT EVEN THINKING ABOUT IT, YOU SIMPLY OBSERVED THAT 25=2∙12+1, AND YOU ADDED 1 MONTH TO SEPTEMBER
- 5. FOR EXAMPLE •If it is now wednesday, you know that in 23 days it will be friday. •This time, you arrived at your answer by noting that 23=3∙ 7+2, so you added 2 days to wednesday instead of counting off 23 days
- 6. WHAT IS MODULAR ARITHMETIC? •Modular arithmetic is an abstraction of a method of counting that you often use. Surprisingly, this simple idea has numerous important applications in mathematics and computer science.
- 7. THE DIVISION ALGORITHM: • Given any positive integer n and any nonnegative integer a, if we divide a by n, we get an integer quotient q and an integer remainder r that obey the following relationship: • a = q n + r 0 r < n ; q = a / n • Where x is the largest integer less than or equal to x. E.G. Let a = 7 and n = 3, then 7 = 2 3 + 1. Here, q = 2 and r = 1, Let a = 10 and n = 2, then 10 = 5 2 + 0. Here, q = 5 and r = 0, Let a = -11 and n = 7, then -11 = (-2) 7 + 3. Here, q = -2 and r = 3.
- 8. WHAT IS MODULAR ARITHMETIC? •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.
- 9. EXAMPLE 3 mod 2 = 1 since 3= 1∙2+1 6 mod 2=0 since 6=3∙2+0, 11 mod 3=2 since 11=3∙3+2, 62 mod 85=62 since 62=0∙85+62, -2 mod 15 = 13 since -2 =(-1)15+13
- 10. a≡ r mod n a: dividend r:remainder n:divisor
- 13. Modular arithmetic operations: • The (mod n) operation maps all integers into the set of integers {0, 1, … , (n -1)}. The modular arithmetic exhibits the following properties:
- 28. How to calculate ab mod n