Problem: 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}
• Explain why the following algorithm that decides LCM does not run in polynomial time:
o Check if m is a multiple of a and b; if not reject a, b, m
o For i = 1, 2, . . . , m - 1 do:
If i is a multiple of a and b, a multiple smaller than m was found.
Reject a, b, m.
o If it reached the end of the loop without finding a multiple less than m, accept a, b, m.
• Prove that LCM ∈ P.
The response should include a reference list. One-inch margins, Using Times New Roman 12 pnt font, double-space and APA style of writing and citations.