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Least common multiples

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Let a and b be integers. A common multiple of a and b is an integer n for which a|n and b|n. 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 [greater than or equal to] m.

The notation for the least common multiple of a and b is lcm(a,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 in Theorem 36.5

(b) Use your formula to show: If a and b are positive integers, then
ab=gcd(a,b)lcm(a,b).

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This shows how to develop a formula for the least common multiple and use the formula in a proof.

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