Least common multiple
Least Common Multiple Example of figuring out the least common multiple of two nunmbers
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- Hello, I'm now going to do some practice least common
- multiple problems for you.
- After I do a couple of these problems you should be able
- to go to the least common multiple module and do
- some of them yourselves.
- Let's say the least common multiple of 10 and 8.
- I'm going to show you two ways to do a least
- common multiple problem.
- One I call just the brute force method and I think it's good
- because it'll give you a good sense of what least common
- multiple is and then I'll also show you what I call the
- more elegant method.
- So the brute force method is literally just to write down
- all the multiples of the two numbers and figure out
- what the least common multiple they have is.
- So let's write all the multiples of 10.
- So 10 times 1 is 10.
- 10 times 2 is 20.
- 30, 40, 50, 60, whoops.
- Not 67.
- 70, 80, 90, 100 and so on.
- Multiples of 8 are 8, 16, 24, 32, 40, 48,
- 64, 72, 80 and so on.
- So let's see.
- Let's see if we can identify what the common multiples are.
- Well, immediately I see that 10 times 4 is 40 and 8 times 5 is
- also 40, so that's a common multiple.
- If we keep going we see that 10 times 8 is 80 and 8
- times 10 is also 80.
- And if we were to keep going we would also see that
- 120 is a common multiple.
- We'd see that 160 is a common multiple.
- But out of the ones we listed 40 and 80 are
- our common multiples.
- And if we were to ask, what is the least common multiple?
- Well, 40 is lower than 80, so we say 40 is the
- least common multiple.
- That's what I call the brute force method.
- Now what I would say the elegant method is, is what you
- do is you look at the factors of 10 and you say, well, the
- factors of 10 are 1, 2, 5, and 10.
- And the factors of 8 are 1, 2, 4, and 8.
- And you say, what's the greatest common factor
- of the two numbers?
- Well, they all share the common factor one.
- Every integer shares that common factor.
- But the number 2.
- They both share that common factor.
- So what we can say is, is that the least common multiple of 10
- and 8-- and this is the elegant way and it might not be obvious
- to you why it works and I might do another module with you
- to show you why this works.
- But the least common multiple of two numbers is always equal
- to the two numbers-- 8 times 10-- and the dot is this
- another fancy way of writing times.
- 8 times 10 and then you divide that by the greatest
- common factor of 8 and 10.
- Well, 8 times 10 is 80, and the greatest common
- factor of 8 and 10?
- Well, we just figured that out.
- That's 2.
- So that equals 40.
- In general, in my head, and you'll learn to do these
- problems, in your head.
- I tend to do it this first way.
- I don't figure out what the greatest common factor is
- and then I multiply the numbers and divide them.
- Because for smaller numbers like 8 or 10 or two and 3, it's
- pretty easy to just think about the multiples and figure out
- the least common multiple.
- But if you had really large numbers or if you're writing a
- computer program, that had to deal with arbitrary numbers,
- then you'd probably want to use the second method.
- And if you're ever in doubt the second method always works just
- to make sure you haven't overlooked some numbers in
- using the method on the left.
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