465 Notes - 8/25 Algorithm sequence of unambiguous...

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8/25 Algorithm – sequence of unambiguous instructions for solving a problem. That is for obtaining a required output from a legitimate input in a finite amount of time [Definition]: x is a divisor of y >x divides y >y is a multiple of x >if exists some k such that y = kx, [x,y,k are integers] [exs]: >1 divides all integers >0 divides only 0 >every integer divides 0 >360 has how many positive integer divisors? >360 = (2 3 )(3 2 )(5 1 ) ># of divisors equals the product of each of the prime factor's exponents plus 1 for the 0 power. ># of divisors = (3+1)(2+1)(1+1) = 24 [Definition]: GCD (greatest common divisor) of integers n and m with not both 0, is max{d|d divides n and d divides m} [exs]: gcd(12, 20) = 4 gcd(1735, 0) = 1735 gcd(0, 0) = undefined but mathematicians state it is 0 Middle School procedure(factoring) [ex]: gcd(420, 550) 420 = (2 2 )*(3 1 )*( 5 1 )*(7 1 ) Factoring algorithm >if one number equals 0, then return other number >Factor m and n >Write m = (P 1 a1 )*(P 2 a2 )*. ..(P
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465 Notes - 8/25 Algorithm sequence of unambiguous...

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