Divisibility Rules 2 to 12
In this section, let us learn about basic divisibility tests from 2 to 12. The divisibility rule of 1 is not required since every number is divisible by 1.
|
Divisibility by number |
Divisibility Rule |
|
Divisible by 2 |
A number that is even
or a number whose last digit is an even number, i.e., 0, 2, 4, 6, and 8. |
|
Divisible by 3 |
The sum of all the digits of the number should be
divisible by 3. |
|
Divisible by 4 |
The number formed by
the last two digits of the number should be divisible by 4 or should be 00. |
|
Divisible by 5 |
Numbers having 0 or 5 as their ones place digit. |
|
Divisible by 6 |
A number that is
divisible by both 2 and 3. |
|
Divisible by 7 |
Subtracting twice the
last digit of the number from the remaining digits gives a multiple of 7. |
|
Divisible by 8 |
The number formed by
the last three digits of the number should be divisible by 8 or should be
000. |
|
Divisible by 9 |
The sum of all the digits of the number should be
divisible by 9. |
|
Divisible by 10 |
Any number whose ones
place digit is 0. |
|
Divisible by 11 |
The difference of the
sums of the alternative digits of a number is divisible by 11. |
|
Divisible by 12 |
A number that is
divisible by both 3 and 4. |
Divisibility Rules Chart and Examples
Let us try to understand the above divisibility
tests with examples.
· Is 280 divisible by 2? Yes, 280 is divisible by 2 as the units place digit is 0.
· Is 345 divisible by 3? Yes, 345 is divisible by 3, as the sum of all the digits, i.e., 3 + 4 + 5 = 12, and 12 is divisible by 3. So, 345 is divisible by 3.
·
Is 450 divisible by 4? No, 450 is not divisible
by 4 as the number formed by the last two digits starting from the right, i.e.,
50 is not divisible by 4.
·
Is 3900 divisible by 5? Yes, 3900 is divisible
by 5 as the digit at the units place is 0 which satisfies the divisibility rule
of 5.
· Is 350 divisible by 6? The sum of all the digits of 350 is 8, so it is not divisible by 3. Hence it cannot be divisible by 6, as a number needs to be a common multiple of both 2 and 3 to be a multiple of 6.
·
357 is divisible by 7 as when we subtract the
twice of the ones place digit, 7 × 2 = 14, and subtract it from the remaining
digits 35, we get 35 -14 = 21, which is divisible by 7. So, 357 is divisible by
7.
·
79238 is not divisible by 8, as the number
formed by the last three digits 238 is not completely divisible by 8.
· 875 is not divisible by 9, as the sum of all the digits, 8 + 7 + 5 = 20 is not divisible by 9.
Divisibility Rule of 11 with Example
The divisibility rule of 11 can also be
understood in a simpler way which says that if the difference between the sums
of the alternate digits of the given number is either 0 or divisible by 11,
then the number is divisible by 11. Let us understand this with an example.
These alternate digits can also be called the digits in the even places and the
digits in the odd places.
Example: Which of the given numbers is exactly
divisible by 11?
a.) 86416
b.) 9780
Solution:
a.) In 86416, if we take the alternate digits
starting from the right, we get 6, 4, and 8 and the remaining alternate digits
are 1 and 6. Now, 6 + 4 + 8 = 18, and 1 + 6 = 7. After finding the difference
between these sums, we get 18 - 7 = 11, which is divisible by 11. Therefore
86416 is divisible by 11. It is to be noted that these alternate digits can
also be considered as the digits on the odd places and the digits on the even
places.
a.) In 9780, if we take the digits on the odd
places, we get 9 and 8 and the digits at the even places are 7 and 0. Now, 9 +
8 = 17, and 7 + 0 = 7. After finding the difference between these sums, we get
17 - 7 = 10, which is neither 0 nor divisible by 11. Therefore 9780 is not
divisible by 11.
Divisibility Rule of 11 For Large Numbers
As we know from the divisibility rule of 11, a
number is divisible by 11 if the difference between the sum of the digits at
the odd and the even places are either equal to 0 or is divisible by 11 without
leaving a remainder. For example, let us find if the number 2541 is divisible
by 11 or not. To check this, let us apply the divisibility test by 11 to the
number 2541. In the number 2541, the digits at the odd positions are 2 and 4
(if we start from the left), hence the sum is 6. The numbers at the even
positions are 5 and 1, hence their sum is 6. Now, the difference between the
sums obtained is 6 - 6, which is equal to 0. We know that 0 is divisible by
every number, so it is divisible by 11. Therefore, the number 2541 is divisible
by 11.
Divisibility Rule of 11 and 12
Divisibility Rules of 11 and 12 are different.
In the divisibility rule of 11, we check to see if the difference between the
sum of the digits at the odd places and the sum of the digits at even places is
equal to 0 or a number that is divisible by 11, whereas the divisibility rule
of 12 states that a number is divisible by 12 if it is completely divisible by
both 3 and 4 without leaving a remainder. Now, let us take a number and check
for the divisibility rule of 11 and 12.
Example: Check the divisibility test of 11 and 12
on the number 764852
Solution: Let us apply the divisibility rule of 11
on this number.
Sum of the digits at odd places (from the left)
= 7 + 4 + 5 = 16
Sum of the digits at even places = 6 + 8 + 2 =
16
Difference between the sum of the digits at odd
and even places = 16 - 16, which is 0.
Therefore, 764852 is divisible by 11.
Let us check if the number is divisible by 12
or not.
For this let us check if the number is
divisible by both 3 and 4. Sum of all the digits = 7 + 6 + 4 + 8 + 5 + 2 = 32.
The Sum of 3 and 2 is 5. 5 cannot be divided by 3 completely. Therefore, 764852
is not divisible by 3. Let us also check the divisibility by 4. For a number to
be divisible by 4, the last two digits of the number should be either '00' or a
number divisible by 4. In the given number, the last two digits are 52. When 52
is divided by 4, the quotient is 13 and the remainder is 0. Hence, we can say
that the number 764852 is divisible by 4.
But for a number to be divisible by 12, it
should pass the divisibility test of 3 as well as 4. Here, we see that the
number is not divisible by 3. So we can say that it is not divisible by 12.
From the example, we can understand that divisibility rules for 11 and 12 are
totally different and it is not necessary that a number that is divisible by 11
should be divisible by 12 also.
