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# Reasoning about unknown variables: divisibility

Video transcript

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.