The why of the 9 divisibility rule

- [Voiceover] Someone walks up to you on the street and says, "2943, quick is this divisible by nine? It's a matter of life and death", and you could say, "Well, I can do this fairly quickly". To figure out whether it's divisible by nine, I just have to add up the digits and figure out if the sum of the digits is a multiple of nine or whether it's divisible by nine. So let's do that. Two plus nine plus four plus three, two plus nine is 11, 11 plus four is 15, 15 plus three is 18, and 18 is definitely divisible by nine, so this is going to be divisible by nine. And even if you are unsure of whether 18 is divisible by nine you could apply the same rule again. One plus eight is equal to nine, so that's definitely, definitely divisible by nine. And so the person can go save their lives or whoever's life they were trying to save with that information. But this might leave you thinking, well, that was nice and useful, but why did that work? Does this work for all numbers? Does it only work for nine? and I don't think this works for eight, or seven, or 11, or 17, why does it work for nine? And actually also works for three, and we'll think about that in the future video. And to realize that, we just have to rewrite 2943. So the two in 2943, it's in the thousands place, so we can literally rewrite it as two times, two times 1000, the nine is in the hundreds place, so we can literally write it as nine times 100, the four is in the tens place, so it's literally the same thing as four times 10, and then finally we have our three in the ones place, we could write it as three times one or just three. This is literally 2000, 900, 40, and three, 2943. But now we can rewrite each of these things, this thousand, this hundred, this ten, as a sum of one plus something that is divisible by nine, so 1000, 1000 I can rewrite as one plus 999, one plus 999, I can rewrite 100 as one plus 99, one plus 99, I can rewrite 10 as one plus 9, and so two times 1000 is the same thing as two times one plus 999, nine times 100 is the same thing as nine times one plus 99, four times 10 is the same thing as four times one plus nine, and then I have this plus three over here. But now I can distribute, I can say, well, this over here is the same thing as two times one, which is just two, plus two times 999, this thing right over here is the same thing as nine times one, just to be clear what I'm doing, I'm distributing the two over the first parenthesis, these first two terms, then the nine, I'm gonna distribute again, so it's gonna be nine, nine times one plus nine times 99, plus nine times 99, and then over here, I forgot the plus sign right over here, I'm gonna distribute the four, four times one, so plus four, and then four times nine, so plus four times nine, and then finally I have this positive three, or plus three right over here. Now I'm just gonna rearrange this addition, so I'm gonna take all the terms we're multiplying by 999, and I'm going to do that in orange. So I'm gonna take this term, this term, and this term right over here, and so I have two times 999, that's that there, plus nine times 99, plus four times nine, plus four times nine, so that's those three terms, and then I have plus two, plus two, plus nine, plus nine, plus four, plus four, and plus three, plus three. And this is interesting, this is just the sum of our digits, this is just what we did up here, and you might see where all of this is going, this orange stuff here, is this divisible by nine? What? Sure, it will definitely, 999, that's divisible by nine, so anything this is multiplying by, it's divisible by nine, so this is divisible by nine, this is definitely divisible by nine, 99, regardless of whether is being multiplied by nine, whatever is multiplying by nine, whatever is multiplying 99 is gonna be divisible by nine, because 99 is divisible by nine, so this works out, and same thing over here, you're always going to be multiplying by a multiple of nine, so all of this business all over here is definitely going to be divisible by nine. And so in order for the whole thing, and all I did is I rewrote 2943 like this right over here, in order for the whole thing to be divisible by nine, this part definitely is divisible by nine, in order for the whole thing does the rest of this sum, it has to be divisible by nine as well. So in order for this whole thing, all of this has to be divisible, divisible by nine.