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