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CCSS.Math: , , ,

let's say that we have three integers a B and C and we know that all of these integers are greater than zero so they're integers and they are greater than zero and we also know that the expression 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 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 multiple multiple of C so this is what we know a B and C are integers all greater than zero we know that the expression A plus B over C also an integer and that a is a multiple of C or in other ways 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 is be a multiple does B have to be a multiple of C let me write it that way does does B given all of these constraints does B have to be have to be a multiple multiple of C so let's see how we can then I encourage you to pause the video right now to kind of come up with your own answer about whether B has to be a multiple of C so now that you've unpause 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 and really one way to tackle this it's 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 over C plus B over C a over C plus B over see and this expression is the exact same thing as our 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 / c we know that a is divisible by c we know that a is a multiple of C so a divided by C this is going to be an integer so let me write that so this information 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 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 this has to be an integer has to be an integer and if B over C is an integer that means that B that yes B must be a multiple of C so the answer here is yes