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# The why of the 9 divisibility rule

## Video transcript

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 multiple but 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 9 is 11 11 plus 4 is 15 15 plus 3 is 18 and 18 is definitely divisible by 9 so this is going to be divisible by 9 and even if you are unsure whether 18 is divisible by 9 you can apply the same rule again 1 plus 8 is equal to 9 so that's definitely definitely divisible by 9 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 9 I don't think this works for 8 or 7 or 11 or 17 why does it work for 9 it actually also works for 3 and we'll think about that in a future video and to realize that we just have to rewrite 2943 so the two and 2943 it's in the thousands place so we can literally write rewrite it as 2 times 2 times 1000 the 9 is in the hundreds place so we can read literally write it as 9 times 100 the 4 is in the tens place so it's literally the same thing as 4 times 10 and then finally we have our 3 in the ones place we could write it as 3 times 1 or just 3 this is this is literally 2940 and 3 2943 but now we can rewrite each of these things this thousand this 100 this 10 as the sum of 1 plus something that is divisible by 9 so a thousand a thousand I can rewrite as 1 plus 999 1 plus 999 I can rewrite a hundred as 1 plus 99 1 plus 99 I can rewrite 10 as 1 plus 9 and so two times a thousand is the same thing as 2 times 1 plus 999 9 times 100 is the same thing as 9 times 1 plus 99 4 times 10 is the same thing as 4 times 1 plus 9 and then I have this plus 3 over here but now I can distribute I can say well this over here is the same thing as 2 times 1 which is just 2 plus 2 times 9 9 9 this thing right over here is the same thing as 9 times 1 just to be clear what I'm doing I'm distributing the 2 on that over the first parenthesis these first two terms then the 9 I'm going to distribute again so it's going to be 9 9 times 1 plus 9 times 99 plus 9 times 99 and then over here I forgot the plus sign right over here I'm going to distribute the 4 4 times 1 so plus 4 and then 4 times 9 so plus 4 times 9 and then finally I have this positive 3 or plus 3 right over here now I'm just going to rearrange this addition so I'm going to take all the terms where I'm 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 2 times 999 that's that there plus 9 times 99 plus 4 times 9 plus 4 times 9 so that's those three terms and then I have plus 2 plus 2 plus 9 plus 9 plus 4 plus 4 and plus 3 plus 3 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 9 well sure it well definitely 999 that's divisible by 9 so anything that's multiplying by it is divisible by 9 so this is divisible by 9 this is definitely divisible by 999 regardless of whether it was being multiplied by 9 whatever is multiplying by not being whatever is multiplying 99 it's gonna be divisible by 9 because 99 is divisible by 9 so this works out and same thing over here you're always going to be multiplying by a multiple of 9 so all of this business right over here is definitely going to be divisible by 9 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 9 this part definitely is divisible by 9 in order for the whole thing does the rest of this sum it has to be divisible by 9 as well so in order for this whole thing all of this has to be divisible divisible by 9