What is the reasoning behind the LCM shortcut?

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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
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Dumb analogy time. You run a bed-and-breakfast, and you have two guests staying this weekend.

The first guest asks for three different kinds of juice. (juice)^(3), if you will.

The second asks for bagels, eggs, and two different kinds of juice. (bagels)x(eggs)x(juice)^(2), if you will.

You have to make a breakfast that will satisfy both of them; what do you make?

The second one wanted bagels, so you have to include bagels. The second one wanted eggs, so you have to include eggs. They both wanted juice – one wanted two kinds, the other wanted three kinds, so if you serve three kinds they’ll both be happy. Your simplest ‘common’ (works for everybody) breakfast is (bagels)x(eggs)x(juice)^3.

That’s what a lowest common multiple is – it’s the simplest number you can come up with that ‘covers’ all the same components as two or more other numbers.

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.