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## 6th grade (Eureka Math/EngageNY)

### Course: 6th grade (Eureka Math/EngageNY) > Unit 2

Lesson 4: Topic D: Number theory—thinking logically about multiplicative arithmetic- Divisibility tests for 2, 3, 4, 5, 6, 9, 10
- Recognizing divisibility
- The why of the 3 divisibility rule
- The why of the 9 divisibility rule
- Divisibility tests
- Intro to even and odd numbers
- Greatest common factor examples
- Greatest common factor explained
- Greatest common factor
- Greatest common factor review
- Least common multiple
- Least common multiple: repeating factors
- Least common multiple of three numbers
- Least common multiple
- Least common multiple review
- GCF & LCM word problems
- GCF & LCM word problems

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# The why of the 9 divisibility rule

Why you can test divisibility by 9 by just adding the digits. Created by Sal Khan.

## Want to join the conversation?

- Maybe I missed this...can you do the same trick where you add the numbers for six since it is divisable by three? Three and nine work. Thanks for any anwsers!(64 votes)
- It doesn’t work for 6 the same way as 3 and 9. If it did, you could say,

18: 1+8 = 9, which is not divisible by 6 even though 18 is. So it doesn't work.

However, as one person suggested but didn’t complete, you can see that if the number were divisible by 2 and 3 then that would make the number divisible by 6. So if the number ends in an even number (0,2,4,6,8) and the digits sum to a number divisible by 3, then the original number is divisible by 6. So for 18:

It ends in 8, which is even, so 18 is divisible by 2.

1+8=9, which is divisible by 3, so 18 is divisible by 3.

Since the number 18 is divisible by both 2 and 3, then it is divisible by 6.(17 votes)

- When figuring out if a number is divisible by 9. Can I just cross out the 9's, cross out numbers that will equal 9 when added (8+1, 7+2, 6+3, 5+4). Then just add the rest? Example: Is 697,225,842 divisible by 9? First crossing out 9's, then the 7 and a 2 because adding them would equal 9. Then doing the same with the 4 and 5. So now I have 6 _ _ , _ 2 _ , 8 _ 2.... Adding these give me 18. YES this is divisible by 9. Can this be a more efficient way to figure out numbers that are divisible by 9?(52 votes)
- Yes, because if you take out numbers that are equal to 9 when added, it will have no effect on the number's overall divisibility by nine.(28 votes)

- why did you think of this and how(7 votes)
- why is it always (for example) 1+99... why is it 99? sure, 1+99=100, but so is 52+48+100(4 votes)
- And because 1 times a number is itself. We want to reach a point where the sum of the digits is isolated and the rest of the numbers are divisible by 9.(1 vote)

- Is six also able to have this rule?(4 votes)
- No for 6 we have to see whether the number is divisible by its factors(2,3) or not.(3 votes)

- I don't understand the distributive form he's doing(4 votes)
- Well, pretend you have 3*(3+5). That is the same thing as3*3+3*5. It's 9+15=24, same as 3+5 (8)*3 (24)(4 votes)

- How i can know how much divisors have a number?? For example the 24 have 8 divisors: 1,2,3,4,6,8,12,24 but how i know it more quickly?(0 votes)
- That's a tough one, trial and error (might want to begin at 2 and work your way up) seems to be the only method. Maybe you could be the next mathematician to invent a revolutionary new way to quickly find the divisors of a number. Best of luck :)(13 votes)

- this trick works only for 3, 6, 9, and on by 3's, right? things divisible by 3. also, can someone explain how to 9's trick for multiplication tables on your fingers works? I

think it's something to do with the fact that if you put one of ten fingers down you always have 9 left.(2 votes)- It's too hard to expain in text. But, there are lots of videos. Copy & past the entire string below into your browser:

https://video.search.yahoo.com/search/video;_ylt=AwrJ7JXXSv5enBYAlkZXNyoA;_ylu=X3oDMTEyZ21uMjRvBGNvbG8DYmYxBHBvcwMyBHZ0aWQDQzAwOTRfMQRzZWMDc2M-?p=multiplication+by+9+using+finger+trick&fr=mcafee&guce_referrer=aHR0cHM6Ly9zZWFyY2gueWFob28uY29tL3NlYXJjaD9mcj1tY2FmZWUmdHlwZT1FMjExVVMxMjc0RzAmcD1tdWx0aXBsaWNhdGlvbitieSs5K3VzaW5nK2Zpbmdlcit0cmljaw&guce_referrer_sig=AQAAAHA_R1wSrof9OPS1EzdUGzCbseKgaVIj1FkpS5_CGYhcghhBoKxEr01MWMNScsOaUbCs_bEx-fxMkJXCi5lyCN8Jm11VobSzlypGFTeLagZ5Zqy0o8m3hhWSiuapHI8GGZYcUBlia8hYDSC5W-QWE_24zSOXlM46XcucF6L5ON-G&_guc_consent_skip=1593723649(2 votes)

- If you could do 3 and 9 (factors of 3) then can you use the same trick on 18? It's a factor of 9!(2 votes)
- Can you use this rule for all multiples of three?(2 votes)

## Video transcript

- [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.