Reasoning about unknown variables: divisibility

Let's say that we have three integers, a, b, and c, and we know that all of these integers are greater than 0. So they're integers, and they are greater than 0. And we also know that the expression a plus b over c, that this is also an integer. The entire expression, if you were to evaluate it, is also an integer. And then finally, we know that a is divisible-- or another way of saying it, that a is a multiple of c, so a is divisible by c, which is another way of saying is a is a multiple of c. So this is what we know. a, b, and c are integers, all greater than 0. We know that the expression a plus b over c is also an integer, and that a is a multiple of c, or a another way is that c divides perfectly into a. So our question for you or the question for all of us to work out right now is, is b a multiple? Does b have to be a multiple of c? Let me write it that way. Does b-- given all of these constraints-- does b have to be a multiple of c? So let's see how we can-- and I encourage you to pause the video right now to come up with your own answer about whether b has to be a multiple of c. So now that you've unpaused things, let's work it out. So let's go to our original expression right over here. We have a plus b over c, and really one way to tackle this, is to really just play around with this expression, and see if we can come up with any conclusions here. So, one, we could try to rewrite a plus b over c. We could rewrite that as a/c plus b/c, and this expression is the exact same thing as our first expression. So we know that this entire thing is going to be an integer. That whole thing is going to be an integer. Now, what do we know about these parts? Well a/c, this is a divided by c. We know that a is divisible by c. We know that a is a multiple of c. So divided by c, this is going to be an integer, so let me write that. So, this information right over here tells us that this thing right over here-- that a divided by c-- is going to be an integer. This is going to be an integer. Now, if I have an integer, and I add something to it, and the whole thing is an integer, well the thing that I'm adding to it must be an integer. The only way that I get an integer plus something to be an integer is if the thing I'm adding it to is also an integer. So this-- there's no way that I could add an integer to a non-integer and then get an integer, so this has to be an integer. And if b/c is an integer, that means that, yes, b must be a multiple of c. So the answer here is yes.