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Modular Arithmetic

Modular Arithmetic. “ Clock” Arithmetic. For integers x and n, x mod n is the remainder of x  n. 0. Examples 7 mod 6 = 1 33 mod 5 = 3 33 mod 6 = 3 51 mod 17 = 0 17 mod 6 = 5. 1. 5. arithmetic mod 6. 2. 4. 3. 11 mod 7 = 4 , since 11 = 7X1 +4.

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Modular Arithmetic

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  1. Modular Arithmetic

  2. “Clock” Arithmetic • For integers x and n, x mod n is the remainder of x  n 0 • Examples • 7 mod 6 = 1 • 33 mod 5 = 3 • 33 mod 6 = 3 • 51 mod 17 = 0 • 17 mod 6 = 5 1 5 arithmetic mod 6 2 4 3

  3. 11 mod 7 = 4, since 11 = 7X1 +4 4 = 7 X(-1) + 11 , therefore4 mod 7 = 11 4 mod 7 = ? -11 mod 7 = ? -11 = 7X(-2) + 3, therefore -11 mod 7 = 3 3 mod 7 = ? 3 = 7X2 -11, therefore 3 mod 7 = -11 a mod n = b  b mod n = a a = b mod n  (a-b) mod n = 0 a = a mod n

  4. Modular Addition • Notation and facts • 7 mod 6 = 1 • 7 = 13 = 1 mod 6 • ((a mod n) + (b mod n)) mod n = (a + b) mod n • ((a mod n)(b mod n)) mod n = ab mod n • Addition Examples • 3 + 5 = 2 mod 6 • 2 + 4 = 0 mod 6 • 3 + 3 = 0 mod 6 • (7 + 12) mod 6 = 19 mod 6 = 1 mod 6 • (7 + 12) mod 6 = (1 + 0) mod 6 = 1 mod 6

  5. Modular Multiplication • Multiplication Examples • 3  4 = 0 mod 6 • 2  4 = 2 mod 6 • 5  5 = 1 mod 6 • (7  4) mod 6 = 28 mod 6 = 4 mod 6 • (7  4) mod 6 = (1  4) mod 6 = 4 mod 6

  6. Modular Inverses • Additive inverse of x mod n, denoted -x, is the number that must be added to x to get 0 mod n • -2 mod 6 = 4, since 2 + 4 = 0 mod 6 • Multiplicative inverse of x mod n, denoted x-1, is the number that must be multiplied by x to get 1 mod n • 3-1 mod 7 = 5, since 3 5 = 1 mod 7

  7. Modular Arithmetic Quiz • Q: What is -3 mod 6? • A: 3 • Q: What is -1 mod 6? • A: 5 • Q: What is 5-1 mod 6? • A: 5 • Q: What is 2-1 mod 6? • A: No number works! • Multiplicative inverse might not exist

  8. Relative Primality • x and y are relatively prime if they have no common factor other than 1 • x-1 mod y exists only when x and y are relatively prime • x-1 mod y is easy to find (when it exists) using the Euclidean Algorithm

  9. Totient Function • (n) is the number of numbers (positive integers) less than n, relatively prime to n • Examples • (4) = 2 since 4 is relatively prime to 3 and 1 • (5) = 4 since 5 is relatively prime to 1,2,3 and 4 • (12) = 4 n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 (n) 1 1 2 2 4 2 6 4 6 4 10 4 12 6

  10. Euler’s theorem If x is relatively prime to n then x(n) = 1 mod n Example: a=3; n=10; (10) = 4; 34= 81  1 mod 10 a=2; n=11; (11) = 10; 310= 1024  1 mod 11 • (p) = p-1 if p is prime • (pq) = (p-1)(q-1) if p and q prime Appendix

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