GCDs & linear (ab)c = a (bc), a a = 0, combinations a + 0 = a, - PowerPoint PPT Presentation

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Arithmetic Assumptions Mathematics for Computer Science MIT 6.042J/18.062J assume usual rules for +, , - : Number Theory: a (b+c) = ab + ac, ab = ba, GCDs & linear (ab)c = a (bc), a a = 0, combinations a + 0 = a, a+1 > a,

Arithmetic Assumptions Mathematics for Computer Science MIT 6.042J/18.062J assume usual rules for +, · · , - : Number Theory: a (b+c) = ab + ac, ab = ba, GCD’s & linear (ab)c = a (bc), a – a = 0, combinations a + 0 = a, a+1 > a, …. Albert R Meyer March 6, 2015 gcd-def. 1 Albert R Meyer March 6, 2015 gcd-def. 2 The Division Theorem Divisibility For b > 0 and any a, have c divides a (c|a) iff q = quotient(a,b) a = k · c for some k � r = remainder(a,b) ∃ unique numbers q, r such that 5|15 because 15 = 3 · 5 a = qb + r and 0 ≤ r < b. n|0 because 0 = 0 · n Take this for granted too! Albert R Meyer March 6, 2015 gcd-def. 3 Albert R Meyer March 6, 2015 gcd-def. 4 1
Simple Divisibility Facts Simple Divisibility Facts • c|a implies c|(sa) • c|a implies c|(sa) [a=k’c implies • if c|a and c|b then (sa)=(sk’)c] c|(a+b) k [if a=k 1 c, b=k 2 c then a+b= (k 1 +k 2 )c ] Albert R Meyer March 6, 2015 gcd-def. 5 Albert R Meyer March 6, 2015 gcd-def. 6 Simple Divisibility Facts Common Divisors c a common divisor of a,b • c|a implies c|(xa) Common divisors of a & b • if c|a and c|b then divide integer linear c|(a+b) (sa+tb) combinations of a & b. integer linear combination of a and b Albert R Meyer March 6, 2015 gcd-def. 7 Albert R Meyer March 6, 2015 gcd-def. 9 2
GCD GCD gcd(a,b) ::= the greatest gcd(a,b) ::= the greatest common divisor of a and b common divisor of a and b gcd(10,12) = 2 lemma: p prime implies gcd(p,a) = 1 or p gcd(13,12) = 1 proof: The only divisors gcd(17,17) = 17 of p are ± 1 & ± p. gcd(0, n) = n for n > 0 Albert R Meyer March 6, 2015 gcd-def. 10 Albert R Meyer March 6, 2015 gcd-def. 11 3
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