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how can you use prime factorization to determine the LCM and GCF of two numbers?

Write down the prime factors of each.

Look at all the factors, of both numbers, together.

The LCM will be the smallest set of this list from which you can make either of the two original numbers.

Example:

30 = 2 * 3 * 5
24 = 2 * 2 * 2 * 3

So we have 2, 2, 2, 2, 3, 3, 5 to choose from, and we need to choose a set from that that will make either of our numbers. That set is: 2 * 2 * 2 * 3 * 5, which is 120.

Now, for the GCF, we need to choose as many of the factors as both number have in common:

30 = 2 * 3 * 5
24 = 2 * 2 * 2 * 3

2 and 3 are in both (only one two, since 30 has only one two), so 2 * 3 = 6 is the GCF.

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