Divisibility tests
The Why of the 3 Divisibility Rule Why you can add the digits to see if something is divisible by 3
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- 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
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