For example two numbers 125 and 150 have prime factors 5^3 and 2x3x5^2. Why is the LCM 2x3x5^3? Why does it work?
In: Mathematics
A multiple of a number has to have at least all the same factors as that number. So a multiple of 125 has to have 2, 3, and 5^2 as factors. Likewise, a multiple of 150 has to have 5^3 as a factor.
A common multiple will have all the factors of the input numbers. So a common multiple of 125 and 150 will have to have 2, 3, 5^2, and 5^3.
A least common multiple will be the smallest number that is a common multiple. We can make our number smaller by removing factors, but the only factors we can remove are the 5^2 because 5^3 contains 5^2 meaning the 5^2 from the 125 is redundant. That leaves 2, 3, and 5^3. We can’t get rid of any more factors without getting rid of a factor from 125 or 150.
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