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## 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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# Divisibility tests for 2, 3, 4, 5, 6, 9, 10

Worked example of basic divisibility tests. Created by Sal Khan.

## Want to join the conversation?

- What about divisibility by 7 and 8? Guess there's no easy way to test that without a calculator ;)(415 votes)
- Actually, divisibility by 7 & 8 is quite easy once you get the hang of it.

First, I will talk about divisibility by 8, since it is easier. In order to test this, you only must check to see whether the last three digits of the number are divisible by 8. If they are, then the entire number is divisible by 8 too.

Example 1: Is the number 8347475537272 divisible by 8?

Answer 1:**Yes**, because the last 3 digits,*272*, are divisible by 8.

Example 2: Is the number 314159265358979323846 divisible by 8?

Answer 2:**No**, because the last 3 digits,*846*, are not divisible by 8.

Next, divisibility by 7. This one is a little weird but it really is quite simple after you practice it a couple of times. In order to test this, you must take the last digit of the number you’re testing and double it. Then, subtract this number from the rest of the remaining digits. If this new number is either 0 or if it’s a number that’s divisible by 7, then then original number is divisible by seven. (You may have to repeat this a couple of times if the divisibility of the resulting number is not immediately obvious).

Example 1: Is the number 364 divisible by 7?

Answer 1:**Yes**: Double the 4 to get 8. Subtract 8 from 36 to get 28. Since 28 is divisible by 7, we can now say for certain that 364 is also divisible by 7.

Example 2: Is the number 8256 divisible by 7?

Answer 2:**No**, Double 6 to get 12. Subtract 12 from 825 to get 813. 813 is slightly too large to tell whether it is divisible by 7 so we must repeat the process. Double 3 to get 6. Subtract 6 from 81 to get 75. Since 75 is not divisible by 7, neither is 813 or 8256. Therefore, 8256 is not divisible by 7.

If I wasn't clear with my explanation or if you need any more help, just ask. I hope that helped. :)

P.S. This edit was made in response to a.ortalda's great question about divisibility by 7. It is as follows:*What about the number used in this video? I tried to test the divisibility by 7 of 2799588, but at a certain point I have 27994 - (2*2) = 27990. How can I continue?*

For this special case, you can just drop the zero from 27990 to 2799 and continue from there. This works because, in essence, you are dividing by 10. Since 10 equals 5*2 (neither of which are 7), it should not influence the result.

To finish the problem, 279 - 9*2 = 261. 26 - 1*2 = 24. Since 24 is not divisible by 7, neither is 2799588.(942 votes)

- Divisibility by 11:

How to check a number is divisible by 11?

It is very simple, check the number, if the difference of the sum of digits at odd places and the sum of its digits at even places, is either 0 or divisible by 11, then clearly the number is divisible by 11.

Examples:

Is the number 2547039 divisible by 11?

First find the difference between sum of its digits at odd and even places.

(Sum of digits at odd places) - (Sum of digits at even places)

= (9 + 0 + 4 + 2) - (3 + 7 + 5)

= 15 - 15 = 0

The number is 0, so the number 2547039 is divisible by 11.

Is the number 13165648 divisible by 11?

(Sum of digits at odd places) - (Sum of digits at even places)

= (8 + 6 + 6 + 3) - (4 + 5 + 1 + 1)

= 23 - 11 = 12

The number is 12, so the number 13165648 is not divisible by 11.

Divisibility by 11:

How to check a number is divisible by 11?

It is very simple, check the number, if the difference of the sum of digits at odd places and the sum of its digits at even places, is either 0 or divisible by 11, then clearly the number is divisible by 11.

Examples:

Is the number 2547039 divisible by 11?

First find the difference between sum of its digits at odd and even places.

(Sum of digits at odd places) - (Sum of digits at even places)

= (9 + 0 + 4 + 2) - (3 + 7 + 5)

= 15 - 15 = 0

The number is 0, so the number 2547039 is divisible by 11.

Is the number 13165648 divisible by 11?

(Sum of digits at odd places) - (Sum of digits at even places)

= (8 + 6 + 6 + 3) - (4 + 5 + 1 + 1)

= 23 - 11 = 12

The number is 12, so the number 13165648 is not divisible by 11.(13 votes) - is there a way to find the multiple of 11 when it's a three digit number? is it possible?(8 votes)
- to find if a number is divisible by 11, find the sum of the first digit, 3, 5, 7... and the sum of the second digit, 4, 6, 8... and see if they are the same number. It does not matter if there are more numbers in one set of digits than another.(4 votes)

- how about divisibility in numbers 7 and 8(5 votes)
- Dividing by 7 (2 Tests)

Take the last digit in a number.

Double and subtract the last digit in your number from the rest of the digits.

Repeat the process for larger numbers.

Example: 357 (Double the 7 to get 14. Subtract 14 from 35 to get 21 which is divisible by 7 and we can now say that 357 is divisible by 7.

NEXT TEST

Take the number and multiply each digit beginning on the right hand side (ones) by 1, 3, 2, 6, 4, 5. Repeat this sequence as necessary

Add the products.

If the sum is divisible by 7 - so is your number.

Example: Is 2016 divisible by 7?

6(1) + 1(3) + 0(2) + 2(6) = 21

21 is divisible by 7 and we can now say that 2016 is also divisible by 7.

Dividing by 8

This one's not as easy, if the last 3 digits are divisible by 8, so is the entire number.

Example: 6008 - The last 3 digits are divisible by 8, therefore, so is 6008.(8 votes)

- what does divisibly mean!(0 votes)
- Divisibility means that a number goes evenly (with no remainder) into a number. For example, 2 goes evenly into 34 so 34 is divisible by 2. But 3 would leave us with a remainder, so 34 is not divisible by 3.(22 votes)

- is there anything as partial divisibility and if there is where can we find its usage? like 90 being divisible by 4 partially because 90/4=22.5(4 votes)
- Divisibility means you can divide and not have a remainder (or a fraction). Yes, you can divide 90 by 4, but you end up with a decimal, which is a fraction. So, you can't say 90 is divisible by 4.(5 votes)

- How to use divisibility tests for 11,13,17,and 19?(6 votes)
- pls salman how about the 7 and 8 important(3 votes)
- The test Sal provided are the ones that are the most useful.

There are rules to test for divisibility by 7. Personally, I think they are too complicated and it is easier to just divide by 7. This site has one of the easier tests for 7: http://www.mathsisfun.com/divisibility-rules.html

The test for 8 is to see if the last 3 digits can be divided by 8. Chances are, that we would need to do long division to do this test. So, why not just divide by 8. Or since 8 = 4 * 2, do the test for 4. If it succeeds, divide by 4. Then do the test for 2.(4 votes)

- whats the divisibility rule for 13(3 votes)
- It's not simple like some of the others, but there is one. Here it is:

To determine if an integer is divisible by 13, remove the last digit, multiply it by 9, and subtract this from the remaining digits. Continue this until you reach a number that you know is divisible or not divisible by 13. If it is divisible by 13, then the original number is divisible by 13, but if it is not divisible by 13, the original number is not. For example, 312 is divisible by 13, because multiplying the last digit by 9 and subtracting it from the rest of the number yields 31 - 9(2) = 13, which is divisible by 13, but 867 is not divisible by 13, because multiplying the last digit by 9 and subtracting it from the rest of the number yields 86 - 9(7) = 23, which is not divisible by 13.(4 votes)

- Why did Khan not include divisibility tests for 7 and 8?(4 votes)
- well, maybe because there is no one for 7, what concerncs 8, that's just 2x4, isn't it?(2 votes)

## Video transcript

What we're going
to do in this video are some real quick tests to see
if these three random numbers are divisible by any
of these numbers here. And I'm not going to focus a
lot on the why of why they're divisible-- we'll do that in
other videos-- but really just to give you a sense
of how do you actually test to see if this is
divisible by 2 or 5 or 9 or 10. So let's get started. So to test whether any of
these are divisible by 2, you really just have to
look at the ones place and see if the ones
place is divisible by 2. And right over here,
8 is divisible by 2, so this thing is going
to be divisible by 2. 0 is considered to
be divisible by 2, so this is going to
be divisible by 2. Another way to think
about it is if you have an even number
over here-- and 0 is considered to be
an even number-- then you're going to
be divisible by 2. And over here, you do not have
a number that is divisible by 2. This is not an even
number, this 5, so this is not divisible by 2. So I won't write any 2 there. So we've gone through the 2s. Now, let's work through the 3s. So to figure out if
you're divisible by 3, you really just have to
add up all the digits and figure out if the
sum is divisible by 3. So let's do that. So if I do 2 plus 7 plus 9
plus 9 plus 5 plus 8 plus 8, what's this going
to be equal to? 2 plus 7 is 9. 9 plus 9 is 18, plus 9 is 27,
plus 5 is 32, plus 8 is 40, plus 8 is 48. And 48 is divisible by 3. But in case you're not sure--
so this is equal to 48-- in case you're not sure whether
it's divisible by 3, you can just add
these digits up again. So 4 plus 8 is equal to 12, and
12 clearly is divisible by 3. And if you're not
even sure there, you could add those
two digits up. 1 plus 2 is equal to 3, and
so this is divisible by 3. This right over here,
let's add up the digits. And we can do this one in
our head pretty easily. 5 plus 6 is 11. 11 plus 7 is 18. 18 plus 0 is 18. And if you want to add the 1
plus 8 on the 18, you get 9. So the digits add up to 9. So these add up to 9. Well, they add to 18, which
is clearly divisible by 3 and by 9, and these two
things will add to 9. So the important
thing to know is when you add up all the digits,
the sum is divisible by 3. So this is divisible by 3
as well, divisible by 3. And then finally, Let's
add up these digits. 1 plus 0 plus 0 plus 7 is 8,
plus 6 is 14, plus 5 is 19. So we summed up the digits. 19 is not divisible by 3. So this one, we're not going
to write a 3 right over there. It's not divisible by 3. Let's try 4. And to think about
4, you just have to look at the last
two digits and to see-- are the last two
digits divisible? Are the last two
digits divisible by 4? Immediately, you can look
at this one right over here, see it's an odd number. If it's not going to
be divisible by 2, it's definitely not going
to be divisible by 4. So this one's not divisible by
any of the first few numbers right over here. But let's think about one, 88. Is that divisible by 4? And you can do
that in your head. That's 4 times 22. So this is divisible by 4. Now, let's see. 4 goes into 60 15 times. And then to go from 60 to 70,
you have to get another 10, which is not divisible by 4. So that's not divisible by 4. And you can even try to
divide it out yourself. 4 goes into 70,
let's see, one time. You subtract, you get a 30. 4 goes into 30 seven times. You multiply, then you subtract. You get a 2 right over
here as your remainder, so it is not divisible by 4. Now, let's move on to 5. Now, you're probably already
very familiar with this. If your final digit is a 5 or
a 0, you are divisible by 5. So this one is not
divisible by 5. This one is divisible by 5. You have a 0 there, so
this is divisible by 5. And this, you have a
5 as your ones digit. So once again-- finally-- this
is divisible by something. It's divisible by 5. Now, the number 6. The simple way to think
about divisibility by 6 is that you have to be
divisible by both 2 and 3 in order to be divisible by 6,
because the prime factorization of 6 is 2 times 3. So here, we're
divisible by 2 and 3, so we're going to be
divisible by-- let me do that in a new
color-- so we're going to be divisible by 6. Here, we're
divisible by 2 and 3, so we're going to
be divisible by 6. And if you were just divisible
by 2 or 3, just one of them, then you wouldn't
be able to do this. You have to have
both a 2 and a 3, divisibility by both of them. And here, you're divisible
by neither 2 nor 3, so you're not going
to be divisible by 6. Now, let's do the test for 9. The test for 9 is very
similar to the test for 3. Sum up all the digits. If that sum is divisible
by 9, then you're there. Well, we already summed
up the digits here, 48. 48 actually is not
divisible by 9. If you're not sure, you can
add up the digits there. You get 12. 12 is definitely
not divisible by 9. So this thing right over
here is not divisible by 9. And this one over here, if
you added up all the digits, we got 18, which is divisible. It is divisible by 9. And I'm running out of colors. So this one is divisible by 9. All the digits added up to 18. And this one over here, you
don't even have to add them up, because we already know
it's not divisible by 3. If it's not divisible by 3,
it can't be divisible by 9. But if you did
add up the digits, you get 19, which is
not divisible by 9. So this also is
not divisible by 9. And then finally,
divisibility by 10. And this is the easiest one
of all, because you just have to see if you have
a 0 in the ones place. You clearly do not have a
0 in the ones place here. You do have a 0 in
the ones place there, so you are divisible by 10 here. And then finally, you don't
have a 0 in the ones place here, so you're not going
to be divisible by 10. Another way you
could think about it, you have to be divisible by both
2 and 5 to be divisible by 10. Here, you are divisible
by 5 but not by 2. But obviously,
the easiest one is to just see if you have
a 0 in the ones place.