  Greatest Common Factor

The Greatest Common Factor, or GCF, is the largest integer that is a factor of two or more integers. Say for example we have three integers in question. So, we’ve got 12, 24, and 6. So, on the other side of these equal signs, we’re going to put its prime factorization down. And so the prime factorization for 12 is 2 times 2 times 3. Prime factorization for 24 would be 2 times 2 times 2 times 3. And 6 is just 2 times 3. Now, what we must look at is what factors, prime factors, that they all three have in common. We notice that we’ve got, all three of them have in common a 2, and we also have everyone also having the same prime factor of 3. So down here for our GCF, all we have to do is write down that, yes, they all have a 2 in  common, and that they all have a 3 in common. So if we multiply that together, our greatest common factor is 6. We can also do this with mathematical expressions. Say for example we have 3x squared times x plus 1, and we have 2x times x plus 1 squared. So once again, on the other side of these equal signs, I’m going to write its prime factorization. So I have 3 times x times x times x plus 1. And on this one, I’ve got 2 times x times x plus 1 times x plus 1. So now all I’m going to do is just circle what they have in common. So, it looks like they have an x in common, and they also have an x plus 1 in common. So now for my GCF, I simply write that down: x times x plus 1. And of course, it’s just x, x plus 1.

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