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Number Theory The Greatest Common Divisor (GCD) R. Inkulu http://www.iitg.ac.in/rinkulu/ (GCD) 1 / 14 Division Algorithm Given integers a and b, with b > 0, there exists unique integers q (quotient) and r (remainder) satisfying a = qb + r with 0 ≤ r < b.1 * the existence was proved using well-ordering property earlier * for the uniqueness part: let a = q0 b + r0 = q00 b + r00 ; then |r0 − r00 | < b and hence |q0 − q00 | < 1 1 extends to b being 6= 0 (GCD) 2 / 14 Few properties of division For integers a, b, c, • a|12 iff a = ±1 • if a|b and b 6= 0, then |a| ≤ |b| • if a|b and a|c, then a|(bx + cy) for arbitrary integers x and y 2 a divides b with remainder 0 is denoted with a|b; when a|b, we say either of these: a is a divisor of b, a is a factor of b, or b is a multiple of a (GCD) 3 / 14 GCD: definition Let a and b be integers, with at least one of them different from 0. The greatest common divisor of a and b, denoted by gcd(a, b), is the positive integer d satisfying: • d|a and d|b, • if c|a and c|b, then c ≤ d. ex. gcd(8, 17) = 1, gcd(−5, 5) = 5, gcd(−8, −36) = 4 (GCD) 4 / 14 Elementary properties of gcd • Let g = gcd(a, b); a = gr, b = gs for integers r and s. Then gcd(r, s) = 1. * direct proof of the contrapositive • Given integers a, b, c, the gcd(a, bc) = 1 iff gcd(a, b) = 1 and gcd(a, c) = 1. * ”⇒” proof by contradiction: gcd(a, bc) ≥ gcd(a, b); analogously, gcd(a, bc) ≥ gcd(a, c) * ”⇐” proof by contradiction: gcd(a, b) (or gcd(a, c)) ≥ prime divisor of gcd(a, bc) (GCD) 5 / 14 gcd(a, b) is a linear combination of a and b For any two integers a and b, not both of which are zero, there exist integers x and y such that gcd(a, b) = ax + by. * the smallest element d in the non-empty set S = {au + bv | au + bv > 0; u, v are integers} is the gcd(a, b) (GCD) 6 / 14 Corollaries to above • If a and b are given integers, not both 0, then the set T = {ax + by|x, y are integers} is precisely the set of all multiples of gcd(a, b). • If a|c and b|c, with gcd(a, b) = 1, then ab|c. (GCD) 7 / 14 Relatively prime: definitions • Two integers a and b, not both of which are 0, are said to be relatively prime (a.k.a. coprime) whenever gcd(a, b) = 1. (GCD) 8 / 14 Relatively prime: definitions • Two integers a and b, not both of which are 0, are said to be relatively prime (a.k.a. coprime) whenever gcd(a, b) = 1. • The integers a1 , a2 , . . . , an are mutually relatively prime if gcd(a1 , a2 , . . . , an ) = 1. • The integers a1 , a2 , . . . , an are pairwise relatively prime if, for each pair of integers ai and aj , gcd(ai , aj ) = 1. (GCD) 8 / 14 Relatively prime: definitions • Two integers a and b, not both of which are 0, are said to be relatively prime (a.k.a. coprime) whenever gcd(a, b) = 1. • The integers a1 , a2 , . . . , an are mutually relatively prime if gcd(a1 , a2 , . . . , an ) = 1. • The integers a1 , a2 , . . . , an are pairwise relatively prime if, for each pair of integers ai and aj , gcd(ai , aj ) = 1. Note that pairwise relatively prime integers must be mutually relatively prime, but the converse is not necessarily true. (GCD) 8 / 14 Few properties related to relative primality • Let a and b be integers, not both zero. Then a and b are relatively prime iff there exist integers x and y such that 1 = ax + by. • Let a and b be integers with gcd(a, b) = d; then gcd( da , db ) = 1. (GCD) 9 / 14 Euclid’s lemma If a|bc, with gcd(a, b) = 1, then a|c. (GCD) 10 / 14 Yet another view of GCD Let a, b be positive integers. For a positive integer d, d = gcd(a, b) iff (i) d|a and d|b, (ii) whenever c|a and c|b, then c|d. (GCD) 11 / 14 LCM: definition The least common multiple of two nonzero integers a and b, denoted by lcm(a, b), is the positive integer m satisfying • a|m and b|m, • if a|c and b|c, with c > 0, then m ≤ c. (GCD) 12 / 14 Relating gcd and lcm For any two positive integers a and b, gcd(a, b) ∗ lcm(a, b) = a ∗ b * m= ab gcd(a,b) (GCD) divides any common multiple c of a and b, hence, m is the lcm(a, b) 13 / 14 Two corollaries • Every common multiple of positive integers a and b is divisible by lcm(a, b). • For any choice of positive integers a and b, lcm(a, b) = ab iff gcd(a, b) = 1. (GCD) 14 / 14