The why of the 3 divisibility rule

You're just walking down the street and someone comes up to you and says "Quick! Quick!-- 4792. Is this divisible by 3? This is an emergency! Tell me as quickly as possible! And luckily you have a little tool in your toolkit where you know how to test for divisibility by 3 Well, you say I can just add up the digits If the sum of that is a multiple of 3 then this whole thing is a multiple of 3 So you say 4 plus 7 plus 9 plus 2 That's 11. Plus 9, it's 20. Plus 2 is 22 That's not divisible by 3 If you're unsure, you can even add the digits of that 2 plus 2 is 4. Clearly not divisible by 3 So this thing right over here is not divisible by 3 And so luckily that emergency was saved But then you walk down the street a little bit more and someone comes up to you--- "Quick! Quick! Quick! 386,802-- Is that divisible by 3?" Well, you employ the same tactic You say, what's 3 plus 8 plus 6 plus 8 plus 0 plus 2? 3 plus 8 is 11. Plus 6 is 17. Plus 8 is 25. Plus 2 is 27 Well, 27 is divisible by 3 And if you're unsure, you could add these digits right over here 2 plus 7 is equal to 9. Clearly divisible by 3 So this is divisible by 3 as well So now you feel pretty good You've helped two perfect strangers with their emergencies You figured out if these numbers were divisible by 3 very very very very quickly But you have a nagging feeling Because you're not quite sure why that worked You've just kind of always known it And so, let's think about why it worked To think about it, I'll just pick a random number But we could do this really for any number But I don't want to go too puffy on it just so you can see it's pretty common sense here And the number we'll use is 498 I can literally use any number in this situation And to think about why this whole little tool this little system works we just have to rewrite 498 We can rewrite the 4- since it's in the hundred's place we can write that as 4 times 100 Or 4 times 100, that's the same thing as 4 times 1 plus 99 That's all this 4 is 400, which is the same thing as 4 times 100 which is the same thing as 4 times 1 plus 99 And the little trick here is I want to write- instead of writing 100, I want to write this as the sum of 1 plus something that is divisible by 3 And 99 is divisible by 3 If I add more digits here- 999, 9999-- they're all divisible by 3 And this is why you can do the same reasoning for divisibility by 9 Because they are divisible by 9 as well Anyway, that's what the 4 in the hundred's place represents This 9 in the ten's place- well that represents 90 or 9 times 10, or 9 times 1 plus 9 And then finally this 8. That's in the one's place 8 times 1, or we just write plus 8 Now we can distribute this 4 This is 4 times 1 plus 4 times 99. So it's 4 plus 4 times 99 Actually let me write it like this. I'm going to write-- Actually let me write it first like 4 plus 4 times 99 Do the same thing over here This is the same thing as plus 9-- do that magenta color- plus 9 plus 9 times 9 And then finally I have this 8 right over here And I can rearrange everything These terms right over here, the 4 times 99, and the 9 times 9 I can write over here 4 times 99- I'll write what's like a different notation plus the 9 times the 9, that's those two terms and then we have the plus 4 plus 9 plus 8 Well, can we now tell whether this is divisible by 3? These terms, these first two terms are definitely divisible by 3 This's divisible by 3 because 99 is divisible by 3 regardless of what we have already you don't even have to look at this This is divisible by 3, so if you're multiplying it it's still going to be divisible by 3 This is divisible by 3, so if you're multiplying this whole thing it's still going to be divisible by 3 If you add two things that are divisible by 3 the whole thing is going to be divisible by 3 So all of this is divisible by 3 And if you have another digit here, you'd done the same exact thing Instead of having 1 plus 99, you'd had 1 plus 999, 1 plus 9999, etc So the only thing you have to really worry about is this part right over here you have to ask yourself in order for this whole thing to be divisible by 3 this part is- well that part is, then this part in order for the whole thing has to be divisible by 3 that also has to be divisible by 3 But what is this right over here? These are just our original digits 498. 4 and 9 and 8 We just have to make sure that when we take the sum it's divisible by 3