lecture11 - Number Theory and Modular Arithmetic God made...

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1 Lecture 11 (October 4, 2011) Number Theory and Modular Arithmetic p-1 p 1 f (pq) = (p-1)(q-1) 0 1 2 3 4 5 6 7 “God made the integers; all else is the work of man.” - Leopold Kronecker, 1823-91. Divisibility: An integer a divides b (written “a|b”) if and only if there exists an integer c such that c*a = b. Primes: A natural number p ≥ 2 such that among all the numbers 1,2…p only 1 and p divide p. Prime numbers are thus “irreducible” Primes: Useful facts 1. If p is a prime and p | ab, then either p | a or p | b (or both). 2. If p ≠ q are two primes, and p | m and q | m, then pq | m. Fundamental Theorem of Arithmetic: Any integer greater than 1 can be uniquely written (up to the ordering of the factors) as a product of prime numbers. Greatest Common Divisor: GCD(x,y) = greatest k 1 s.t. k|x and k|y. Least Common Multiple: LCM(x,y) = smallest k 1 s.t. x|k and y|k.
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2 You can use MAX(a,b) + MIN(a,b) = a+b applied appropriately to the factorizations of x and y to prove the above fact… Fact: GCD(x,y) × LCM(x,y) = x × y (a mod n) means the remainder when a is divided by n. a mod n = r a = dn + r for some integer d Definition: Modular equivalence a b [mod n] (a mod n) = (b mod n) n | (a-b) a = q n + b for some q Written as a n b, and spoken “a and b are equivalent or congruent modulo n” 31 81 [mod 2] 31 2 81 31 80 [mod 7] 31 7 80 n is an equivalence relation In other words, it is Reflexive: a n a Symmetric: (a n b) (b n a) Transitive: (a n b and b n c) (a n c) n induces a natural partition of the integers into n “residue” classes. (“residue” = what’s left over = “remainder”) Define residue class [k] = the set of all integers that are congruent to k modulo n . Residue Classes Mod 3: [0] = { …, -6, -3, 0, 3, 6, . .} [1] = { …, -5, -2, 1, 4, 7, . .} [2] = { …, -4, -1, 2, 5, 8, . .} [- 6] = { …, -6, -3, 0, 3, 6, . .} [7] = { …, -5, -2, 1, 4, 7, . .} [- 1] = { …, -4, -1, 2, 5, 8, . .} = [0] = [1] = [2]
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3 Why do we care about these residue classes? Because we can replace any member of a residue class with another member when doing addition or multiplication mod n and the answer will not change To calculate: 249 * 504 mod 251 just do -2 * 2 = -4 = 247 We also care about it because computers do arithmetic modulo n, where n is 2^32 or 2^64. Fundamental lemma of plus and times mod n : If (x n y) and (a n b). Then 1) x + a n y + b 2) x * a n y * b Proof of 2: xa = yb (mod n) (The other proof is similar…) x n y iff x = i n + y for some integer i a n b iff a = j n + b for some integer j xa = (i n + y)(j n + b) = n(ijn+ib+jy) + yb n yb Another Simple Fact: If (x n y) and (k|n), then: x k y Example: 10 6 16 10 3 16 Proof: x n y iff x = in + y for some integer i Let j=n/k, or n=jk Then we have: x = ijk + y x = (ij)k + y therefore x k y A Unique Representation System Modulo n:
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This note was uploaded on 11/03/2011 for the course CS 251 taught by Professor Gupta during the Spring '11 term at Carnegie Mellon.

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lecture11 - Number Theory and Modular Arithmetic God made...

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