Time complexity of the least common multiple (lcm) algorithm

Given two numbers a and b, the least common multiple (lcm) of a and b is the smallest number m such that both a and b are factors of m. For example, lcm(15, 21) = 105 because it is the smallest number that has both 15 and 21 as factors.

Formally, we will work with the following decision problem:

LCM = {a, b, m | lcm(a, b) = m}

Save your time - order a paper!

Get your paper written from scratch within the tight deadline. Our service is a reliable solution to all your troubles. Place an order on any task and we will take care of it. You won’t have to worry about the quality and deadlines

Order Paper Now

(a) Explain why the following algorithm that decides LCM does not run in polynomial time:

a) Check if m is a multiple of a and b; if not reject a, b, m

b) For i = 1, 2, . . . , m − 1 do:

i. If i is a multiple of a and b, a multiple smaller than m was found.

Reject a, b, m.

c) If it reached the end of the loop without finding a multiple less than m, accept a, b, m.

(b) Prove that LCM ∈ P.