find the LCM and HCF of 426 and 576 by applying prime factorization method

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Matthew Elliott

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find the LCM and HCF of 426 and 576 by applying prime factorization method

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We're asked to find the least common multiple highest common factor of four. and 5 76. Alright, By using the Factory Ization method. So, we have to go ahead and and break out the factors of this. So, I can see that six goes into this because six goes into 42 7 times uh and then bring down the six. So 71 would be six times 71. And I believe that 71 is actually prime 71. Yeah, so 71 is prime. So we can still break down the six into a two times three, but that's really going to be it. So, the prime factor Ization for this is two times three times 71 for 5, 76. Well, that's a perfect square. That is 24 times 20 for um And for 24 times 24. Let's see. We can make that one a six times a four. And this 16 times a four as well. Alright. And then this is three times two and this is two times two. And these are my primes here. And this one is also a three times a two and a two times a two. All right. A lot more factors to deal with in that one. So, let's see uh This is 123456. So, two to the sixth, times three squared. It's never a bad idea to just take a second when you do one of these factories Ations and and make sure that we're getting what we would expect. Um So, clear this out here. Two raised to the 6th. Times three raised to the second. We do get 576. Alright, well the nice thing is from there, it's easy to answer these two questions. The least common multiple. Alright. The least common multiple. Well, we need the highest degree of all of the factors that we have. All right, for the least common multiple to make sure that this um you know for 26 goes into it. I need to make sure there's a two to the first but in order to make sure 576 goes into it, I need a two to the 6th. So I'll take the highest of the powers. This one has a three to the first. This one has a three squared, so three squared we have to take there. And this one has a 71. This one doesn't at all. But we still need that 71 to make sure that this will go evenly into it. Alright, so that is how we're gonna find our at least common multiple. So let's calculate what that is. Two raised to the sixth uh times nine and then times 71. And that's going to be 40896896 of 40896. That's the smallest number that both of those two original numbers divide evenly into. Alright. For the highest common factor, highest common factor. That's exactly the opposite we have to take out the smallest of all of the powers that we have that they actually share. So this one has a two to the first but this one has a two to the six. So I can only take out the two to the first. This one has a three. This one has a three square, so I can only take out the three. Uh This one has a 71. This one does not. So really that's it, it's going to be just two times three is six. For your highest common factor. Alright. So that's how we find them using the individual factors. For the least common multiple, we take the highest of all of the factors and for the highest common factor we take the smallest of the ones that they share.

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