Assignment:
Let a and b be integers. A common multiple of a and b is an integer n for which amn and b|n. We call an integer m the least common multiple of it provided (1) m is positive, (2) in is a common multiple of a and b, and (3) if n is any other positive common multiple of a and b, then n > m.
The notation for the least common multiple of a and b is lc*, b). For example, lcm(24, 30) = 120.
Please do the following:
(a) Develop a formula for the least common multiple of two positive integers in terms of their prime factorizations; your formula should be similar to the one in Theorem.
(b) Use your formula to show: If a and b are positive integers, then
ab = gcd(a, b) lcm(a, b).
Provide complete and step by step solution for the question and show calculations and use formulas.