CONNECT: Divisibility
If a number can be exactly divided by a second number, with no remainder,
then we say that the first number is divisible by the second number. For
example, 6 can be divided by 3 so we say that 6 is divisible by 3. (We can
also say that 3 is a factor of 6 and 6 is a multiple of 3, but that is not what this
handout is about!)
It can be difficult to divide numbers which are large or perhaps unfamiliar to
you. However, there are procedures that enable us to quickly work out
whether a number is divisible by other numbers. (The methods don’t give the
answer for the division problem, they are simply a guide to see if the number
CAN be divided by the other number.) This is useful for simplifying fractions,
or in a division problem to check if your answer should have a remainder or
not.
You need to know that the word digit refers to symbols for numbers: 0, 1, 2, 3,
4, 5, 6, 7, 8, and 9. For example, the number 453 872 contains the digits 2, 3,
4, 5, 7 and 8. We can add the value of each of those digits together or do
other maths operations on them. So, if we add all the digits in 453 872, we get
4+5+3+8+7+2, and that gives 29. If we add the digits in 1253226, we get
1+2+5+3+2+2+6, which is 21.
Divisibility Tests
2
If a number is even (that is, if its last digit is 2, 4, 6, 8, or 0), it can
definitely be divided by 2. For a number that is as large as 3 694 806 it is now
easy to see that we can divide it exactly by 2 (because it ends in 6).
3
To work out whether a number can be divided by 3, add up all the digits
in the number. If that total is divisible by 3, then the original number is also
divisible by 3. Our example above, 3 694 806, can be divided by 3, because
when you add its digits, you get 36 (3+6+9+4+8+0+6=36) which can be
exactly divided by 3. (To calculate what the value is when 3 694 806 is divided
by 3, we then need to do the division procedure, but at least we know there
should not be any remainder!)
4
To work out whether a number can be divided by 4, if its last two digits,
when read as a two-digit number, can be divided by 4, then the whole number
can be divided by 4. For example, we know straight away that 3 257 816 CAN
be divided by 4 because 16 can! A fast way to divide by 4 is to divide the
1
original number by 2, and then by 2 again. (You should get 814454 when you
divide 3 257 816 by 4). For our example above, however, 3 694 806, we see
that the last two digits together are 06 (which is the same as 6) and 6
CANNOT be exactly divided by 4, so don’t bother trying to divide 4 into it (or, if
you do, you should get a remainder!).
5
If the number you are trying to divide ends with 0 or 5, then it is divisible
by 5. If it ends in anything else, you will have a remainder when you divide.
So, we can immediately tell that 489 124 665 is divisible by 5. Of course we
would then have to do the division to obtain the answer, but at least we know
there will be no remainder when we divide.
6
If a number can be divided by both 2 and 3, then it is divisible by 6. So,
if a number is even and if the total of its digits can be divided by 3, then the
original number is divisible by 6. This means that our number from above,
3 694 806, MUST be divisible by 6 because it is both even and the sum of its
digits (36) is divisible by 3. Let’s do the division:
61 5 801
6)3 693 44 806
(Note that there is no remainder. So the
rule, that the number is both even and
divisible by 3, works.)
7
Unfortunately 7 is not so straightforward, however, give this a try!
To work out if a number is divisible by 7, take the last digit off the number,
double that digit and subtract the doubled number from the remaining number.
If the result is evenly divisible by 7 – it may be negative – then the number is
divisible by seven. This may need to be repeated several times.
Example: Is 896 divisible by 7?
Take off the last digit (6). Double this (12) and subtract it from 89.
This gives 77. Now, 77 is divisible by 7 so the original number,
896 is also divisible by 7. (With 896, it might have been just as
quick to actually divide by 7!)
Now try 37 443. Is 37 443 divisible by 7?
Take off the last digit, 3. Double it (6) and subtract 6 from 3744.
2
896, last
digit 6,
double 6
gives 12:
89–
12
77
3 744 –
6
3 738
Remove the 8, double it (16) and take this away from 373.
373 –
16
357
Remove the 7, double it (14) and take this away from 35, gives 21 which is
divisible by 7. (PHEW!)
35 –
14
21
This means that 37 443 is divisible by 7. But as you can see, it may have
been easier to just divide in the first place!
To find the result, we DO need to divide, of course. But we know we will not
get a remainder, at least!
To divide, we would do:
8
534 9
7)372 43 46 3
If the number formed by the last three digits of the number is divisible by
8, then the whole number is divisible by 8. This is not the easiest rule to apply
however as you need to be able to divide the last three digits by 8 – you may
as well just divide the complete number by 8! BUT, if you divide the number
by 2, then by 2 again, then by 2 again, (that is, you halve the number, then
halve THAT number, then you halve THAT number!) you have actually
divided it by 8! For example, the number from above, 3 694 806 is NOT
divisible by 8 because 806 is not divisible by 8. (Try halving, halving and
halving – you won’t be able to halve even a second time because you get 403
on the first go.)
9
9 is similar to 3. Add the digits of the number and if that total is divisible
by 9 then the number itself is also divisible by 9. Our example, 3 694 806, is
divisible by 9 because the sum of its digits is 36, which is divisible by 9.
3
10
If a number ends with 0 then it is divisible by 10. For example, 67 890
is divisible by 10, but 67 891 is not.
11
Add the digits in the odd positions of the number; add the digits in the
even positions in the number. Subtract the two sums. If the result is 0 or is
divisible by 11, then the original number is divisible by 11.
Example: Is 53 834 divisible by 11?
Add the digits in the odd positions: 4 + 8 + 5 = 17.
Add the digits in the even positions: 3 + 3 = 6.
Subtract: 17 – 6 = 11, which is divisible by 11.
(The answer when you divide 53 834 by 11 is 4894.)
12
Is the number divisible by 4? Is the number divisible by 3? If the
answer to both questions is “yes” then the number is also divisible by 12.
Example: 3 694 806 is NOT divisible by 12 because it is not divisible by 4
(from the rule above). 3 257 816 is NOT divisible by 12 because it is not
divisible by 3 (add the digits, you will get 32). But 3 257 820 IS divisible by 12
because it is divisible by both 3 (add the digits to get 27) and 4 (the last 2
digits, 20, make a number that is divisible by 4). That means we should get
no remainder when we divide by 12:
2 714 8 5
12)328 51 7 5 810 26 0
So, 3 257 820 is divisible by 12 with the result of 271 485
I have summarised the divisibility tests on the next page and there are some
examples for you to try after that:
4
Divisibility tests - summary
Number to Rule
divide by
2
If the number is even (ends in 2, 4, 6, 8, 0), it is divisible by 2.
3
If the total of the digits is divisible by 3, then the number is
divisible by 3.
4
If the last two digits of the number form a number which is
divisible by 4, then the original number is divisible by 4.
5
If the number ends in 0 or 5, then it is divisible by 5.
6
If the number is even and is also divisible by 3, then it is divisible
by 6.
7
Take the last digit off. Double it. Subtract it from the number
remaining. Is the result divisible by 7? If yes, the original
number is divisible by 7, if not, the original number is not divisible
by 7. Can’t tell? Repeat the process with the new result.
8
If the number formed by the last 3 digits is divisible by 8, then the
original number is divisible by 8.
9
If the total of the digits is divisible by 9, then the number is
divisible by 9.
10
If the number ends in 0 it is divisible by 10.
11
Add the digits in the odd positions in the number; add the digits
in the even positions in the number. Subtract the two sums. If
the result is 0 or is divisible by 11, then the original number is
divisible by 11.
12
If the number is divisible by both 3 and 4, then the number is
also divisible by 12.
5
Exercises
Complete the following divisibility table for 4620.
4620 is divisible
by:
Yes or No
2
5
8
9
11
12
Yes
Complete the following divisibility table for 180 473
180 473 is
divisible by:
Yes or No
3
4
7
9
11
Complete the following divisibility table for 13 320.
4620 is divisible
by:
Yes or No
2
5
8
9
11
12
Yes
Complete the following divisibility table for 386 250
180 473 is
divisible by:
Yes or No
3
4
(Solutions are on the next page)
6
5
7
9
11
12
Solutions
Complete the following divisibility table for 4620.
4620 is divisible
by:
Yes or No
2
5
8
9
11
12
Yes
Yes
No
No
Yes
Yes
Complete the following divisibility table for 180 473
180 473 is
divisible by:
Yes or No
3
4
7
9
11
No
No
No
No
No
Complete the following divisibility table for 13 320.
4620 is divisible
by:
Yes or No
2
5
8
9
11
12
Yes
Yes
Yes
Yes
No
Yes
Complete the following divisibility table for 386 250
386 250 is
divisible by:
Yes or No
7
3
4
5
7
9
11
12
Yes
No
Yes
No
No
No
No