Let a and b be integers. A common multiple of a and b is an integer n for which ajn and
Chapter 39, Problem 39.10(choose chapter or problem)
Let a and b be integers. A common multiple of a and b is an integer n for which ajn and bjn. We call an integer m the least common multiple of n provided (1) m is positive, (2) m 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 lcm.a; b/. For example, lcm.24; 30/ D 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 39.5. b. Use your formula to show that if a and b are positive integers, then ab D gcd.a; b/lcm.a; b/:
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