Find the GCF of 56𝑎𝑏 and 46𝑏, given that 𝑎 and 𝑏 are positive, whole numbers.
GCF stands for greatest common factor. So what we need to do is to take 56𝑎𝑏 and 46𝑏 and separate it into its prime factors. Let’s begin with 56𝑎𝑏. 56 is eight times seven. Seven is a prime number. So we can leave it alone. But what does prime mean?
To be prime means that it will have exactly two factors, one and itself. So here, seven can only be broken down into one times seven, one and itself. However, eight is four times two, where two is prime. So we can leave that alone. But four is two times two. So we’ve broken 56 down to its prime factors. 𝑎 is simply 𝑎 and 𝑏 is simply 𝑏.
Now, if they had been something like 𝑎 squared 𝑏, we could see that 𝑎 squared could be broken down into 𝑎 times 𝑎. But we don’t have that. So 56𝑎𝑏 could be rewritten as two times two times two times seven times 𝑎 times 𝑏. And if we wanted to, we could rewrite two times two times two as two cubed.
Now, let’s take a look at 46𝑏. 46 is two times 23, where two is prime and 23 is prime. And then, 𝑏 is all by itself. So we can rewrite 46𝑏 as two times 23 times 𝑏. So now, we’re asked to find the greatest common factor. So what is the factor that’s the greatest that they have in common? Well, comparing these two numbers, they each have a two and a 𝑏. Therefore, the greatest common factor between these two numbers would be two 𝑏.
Now, if, for example, we had the same problem but we have this extra two over here, then they would have these twos in common and these twos in common. So you would take two times the other set of twos that they have in common and multiply them together to be four. And then, they each would have a 𝑏. So the greatest common factor would be four 𝑏.
However, in our case, our final answer will be two 𝑏.