Divisibility rules, help the students to find if a number is divisible by another number without doing lengthy calculations.
Meaning of ‘Divisible by’: ‘Divisible by’ means when we divide a number with another number, the result is a whole number, means division is a complete division (remainder is zero).
Rules:
Note: Zero (0) is divisible by any number except by itself means 0 can be divided by any number but \(0 ÷ 0\) is meaningless.
Divisibility by 1: Every integer is divisible by 1.
Divisibility by 2
: any integer whose last digit is even means any number ending with 0, 2, 4, 6, 8, is divisible by 2.
Example: look at the number 198, last digit in this number is 8, therefore 198 is divisible by 2.
Some more examples for the divisibility by 2:
| Number |
Divisible by 2 |
Reason |
| 32 |
Yes |
Ends with 0 |
| 332 |
Yes |
Ends with 2 |
| 1554 |
Yes |
Ends with 4 |
| 16676 |
Yes |
Ends with 6 |
| 27 |
No |
Ends with 7 |
| 123 |
No |
Ends with 3 |
Divisibility by 3
: A number is divisible by 3 if the sum of the digits of the number is a multiple of 3, means if the sum of digits is divisible by 3.
Example: Suppose we need to check if the number 72 is divisible by 3 or not
The digits of 72 are 7 and 2 sum of digits = \(7+2=9\)
9 is a multiple of 3 as \(9÷3=3\)
So, 72 is divisible by 3
Some more examples for the divisibility by 3:
| Number |
Divisible by 3 |
Reason |
| 105 |
Yes |
sum of digits \(1+0+5=6\) which is divisible by 3 |
| 4314 |
Yes |
sum of digits \(4+3+1+4=12\) which is divisible by 3 |
| 577 |
No |
sum of digits \(5+7+7=19\) which is NOT divisible by 3 |
| 71 |
No |
sum of digits \(7+1=8\) which is NOT divisible by 3 |
| 695 |
No |
sum of digits \(6+9+5=20\) which is NOT divisible by 3 |
| 423 |
Yes |
sum of digits \(4+2+3=9\) which is divisible by 3 |
Note: This rule can be repeated when needed, look at the following example:
Consider a number 89988
sum of digits of \(89988=8+9+9+8+8=42\)
now,reapeat the rule for number 42
sum of digits of \(42=4+2=6\)
and,6 is divisible by 3 so,89988 is also divisible by 3.
Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4 means number formed by its last two digits is a multiple of 4.
Example: Suppose we need to check if the number 172 is divisible by 4 or not
Now, the number formed by last two digits of 172 is 72
divide 72 with 4
example: The number 72 is a divisible by 4 as \(72÷4=18\)
So the number 172 is divisible by 4
Some more examples for the divisibility by 4:
| Number |
Divisible by 4 |
Reason |
| 2324 |
Yes |
Number formed by last two digits (24) is divisible by 4 as \(24÷4=6\) |
| 3860 |
Yes |
Number formed by last two digits (60) is divisible by 4 as \(60÷4=15\) |
| 175 |
No |
Number formed by last two digits (75) is NOT divisible by 4 |
| 2461 |
No |
Number formed by last two digits is NOT divisible by 4 |
| 695 |
No |
Number formed by last two digits (95) is NOT divisible by 4 |
| 15664 |
Yes |
Number formed by last two digits (64) is divisible by 4 |
Divisibility by 5: any integer ending with 0 or 5 means any integer which has las digit 0 or 5, is divisible by 5.
Example: Suppose we need to check if the number 90 is divisible by 5 or not
The last digit in 90 is 0, so 90 is divisible by 5.
More examples for the divisibility by 5:
| Number |
Divisible by 5 |
Reason |
| 135 |
Yes |
Ends with 5 |
| 3860 |
Yes |
Ends with 0 |
| 15875 |
Yes |
Ends with 5 |
| 2460 |
Yes |
Ends with 0 |
| 697 |
No |
Ends with 7 |
| 15664 |
No |
Ends with 4 |
Divisibility by 6: Any integer which is divisible by both 2 and 3.
Example: Suppose we need to check if the number 162 is divisible by 6 or not
Check divisibility by 2: As the last digit of number 162 is 2, so 162 divisible by 2 (by the
divisibility rule of 2).
Check divisibility by 3: sum of digits of 162=1+6+2=9, now 9 is divisible by 3, so 162 is also divisible by 3 (by the
divisibility rule of 3).
Now we can see that the number 162 is divisible by both 2 and 3, so it’s also divisible by 6.
Another example:Suppose we need to check if the number 274 is divisible by 6 or not
Check divisibility by 2: As the last digit of number 274 is 4, so 274 divisible by 2 (by the
divisibility rule of 2).
Check divisibility by 3: sum of digits of 274\(=2+7+4=13\), now 13 is not divisible by 3 (by the
divisibility rule of 3), so 274 is not divisible by 3.
Now we can see that the number 274 is divisible by 2 but not by 3, so it’s not divisible by 6.
Another example: Suppose we need to check if the number 165 is divisible by 6 or not
Check divisibility by 2: As the last digit of number 165 is 5, so 165 is not divisible by 2 (by the
divisibility rule of 2).
Check divisibility by 3: sum of digits of 165=\(1+6+5=12\), now 12 is divisible by 3 (by the
divisibility rule of 3) so, 165 is also divisible by 3
Now we can see that the number 165 is not divisible by 2 but it’s divisible by 3, so it’s not divisible by 6.
Divisibility by 7: To check if a number is divisible by 7, double the last digit and subtract from the number formed by rest of the digits, if the result is a multiple of/divisible by 7 then the number is also divisible by 7.
Example: Suppose we need to check if the number 91 is divisible by 7 or not
Now, last digit of 91 is 1, when we double it means multiply 1 by 2 and we will get 2
And, number formed by rest of the digits is 9
Subtract 2 from 9 and we will get 7 as \(9-2=7\), which is divisible by 7, so 91 is divisible by 7.
Another example: Suppose we need to check if the number 682 is divisible by 7 or not
Now, last digit of 682 is 2, when we double it means multiply 2 by 2 and we will get 4
And, number formed by rest of the digits is 68
Subtract 4 from 68 and we will get 64 as \(68-4=64\), which is not divisible by/not a multiple of 7, so 682 is not divisible by 7.
Note: This rule can be repeated when needed, look at following example:
Suppose we need to check if the number 1771 is divisible by 7 or not
Now, last digit of 1771 is 1, when we double it means multiply 1 by 2 and we will get 2
And, number formed by rest of the digits is 177
Subtract 2 from 177 and we will get 175 as \(177-2=175,\)
Repeat the rule for 175:
Now, last digit of 175 is 5, when we double it means multiply 5 by 2 and we will get 10
And, number formed by rest of the digits is 17
Subtract 10 from 17 and we will get 7 as \(17-10=7\), which is divisible by 7, so 1771 is divisible by 7.
Divisibility by 8: A number is divisible by 8 if the number formed by its last three digits is divisible by 8 means number formed by its last three digits is a multiple of 8.
Example: Suppose we need to check if the number 3120 is divisible by 8 or not
Now, the number formed by last three digits of 3120 is 120
now divide 120 with 8
the number 120 is divisible by 8 as \(120÷8=15\)
so the number 3120 is divisible by 8.
More examples for the divisibility by 8:
| Number |
Divisible by 8 |
Reason |
| 2824 |
Yes |
Number formed by last three digits is divisible by 8;
\((824÷8=103)\)
|
| 53104 |
Yes |
Number formed by last three digits is divisible by 8;
\((104÷8=6638)\)
|
| 1166 |
No |
Number formed by last three digits is divisible by 8; (166 is not divisible by 8) |
| 12461 |
No |
Number formed by last three digits is divisible by 8; (461 is not divisible by 8) |
| 15664 |
Yes |
Number formed by last three digits is divisible by 8;\((664÷8=83)\) |
Divisibility by 9: A number is divisible by 9 if the sum of the digits of the number is a multiple of 9, means if the sum of digits is divisible by 9.
Example: Suppose we need to check if the number 3672 is divisible by 9 or not
The digits of 3672 are 3, 6, 7 and 2 sum of digits = \(3+6+7+2=18\)
18 is a multiple of 9 as \(18÷9=2\)
So, 3672 is divisible by 9
More examples for the divisibility by 9:
| Number |
Divisible by 9 |
Reason |
| 135 |
Yes |
sum of digits\(=1+3+5=9\)
and 9 is divisible by 9
|
| 9846 |
Yes |
sum of digits\(=9+8+4+6=27\)
and 27 is divisible by 9
|
| 577 |
No |
sum of digits\(=5+7+7=19\)
and 19 is not divisible of 9 |
| 74 |
No |
sum of digits\(=7+4=11\)
and 8 is not divisible by 9
|
Note: This rule can be repeated when needed, look at the following example:
Consider a number 16299
sum of digits of 16299\(=1+6+2+9+9=27\)
reapeat the rule for number 27
sum of digits of \(27=2+7=9\)
and,9 is divisible by 9 so,16299 is also divisible by 9.
Divisibility by 10: any integer ending with 0, is divisible by 10.
Example: Suppose we need to check if the number 170 is divisible by 10 or not
Last digit of number 170 is 0, so 170 is divisible by 10
Divisibility by 11: To check if a number is divisible by 11, subtract the last digit from the number formed by rest of the digits, if the result is a multiple of/divisible by 11, then the number is also divisible by 11.
Example: Suppose we need to check if the number 297 is divisible by 11 or not
Last digit of 297 is 7
Number formed by rest of the digits = 29
Now, subtract 7 from 29 and the result will be 29-7=22, which is a multiple of 11 (22 is divisible by 11), so 297 is divisible by 11.
Note: This rule can be repeated when needed, look at the following example:
Suppose we need to check if the number 14652 is divisible by 11 or not
Last digit of 14652 is 2
Number formed by rest of the digits = 1465
Now, subtract 2 from 1465 and the result will be \(1465-2=1463\),
Now, repeat the rule for 1463:
Last digit of 1463 is 3
Number formed by rest of the digits = 146
Now, subtract 3 from 146 and the result will be \(146-3=143\),
Again, repeat the rule for 143:
Last digit of 143 is 3
Number formed by rest of the digits = 14
Now, subtract 3 from 14 and the result will be \(14-3=11\), which is a multiple of 11 (11 is divisible by 11), so 14652 is divisible by 11.
Another example: Suppose we need to check if the number 976 is divisible by 11 or not
Last digit of 976 is 6
Number formed by rest of the digits = 97
Now, subtract 6 from 97 and the result will be \(97-6=91\), which is not a multiple of 11 (91 is not divisible by 11) so, 976 is not divisible by 11.
Another Rule to Check Divisibility By 11:
If in any integer, the sum of the digits at even position = the sum of the digits at odd position, then the number is divisible by 11.
OR if, the difference of the sum of the digits at even position and the sum of the digits at odd position =11, then the number is divisible by 11.
Example: Suppose we need to check if the number 1364 is divisible by 11 or not
Now, digits at Odd position are 4 and 3
sum of 4 and 3\(=4+3=7\)
Digits at Even position are 6 and 1
sum of 6 and 1\(=6+1=7\)
Sum of digits at Odd position = Sum of digits at Even position = 7
So, 1364 is divisible by 11.
Another example: Suppose we need to check if the number 1903 is divisible by 11 or not
Now, digits at Odd position are 3 and 9
sum of 3 and 9\(=3+9=12\)
Digits at Even position is 1 and 0
sum of 1 and 0\(=1+0=1\)
Difference of Sum of digits at Odd position and Sum of digits at Even position \(= 12-1=11\)
So, 1903 is divisible by 11.
Divisibility by 12: Any integer that is divisible by both 3 and 4 is also divisible by 12.
Example: Suppose we need to check if the number 660 is divisible by 12 or not
Check divisibility by 3: sum of digits of 660\(=6+6+0=12\), now 12 is divisible by 3 (because 12 is a multiple of 3 as \(12÷3=4)\) so, 660 is also divisible by 3.
Check divisibility by 4: number formed by last two digits of 660 is 60, now 60 is divisible by 4 (because 60 is a multiple of 4 as \(60÷4=15)\) so, 660 is also divisible by 4
Now we can see that the number 660 is divisible by both 3 and 4, so it’s also divisible by 12.
Another example: Suppose we need to check if the number 648 is divisible by 12 or not
Check divisibility by 3: sum of digits of \( 548=6+4+8=18\), now 18 is divisible by 3 (because 18 is a multiple of 3 as \(18÷3=6)\) so, 648 is also divisible by 3.
Check divisibility by 4: number formed by last two digits of 648 is 48, now 48 is divisible by 4 (because 48 is a multiple of 4 as \(48÷4=12)\) so, 648 is also divisible by 4
Now we can see that the number 648 is divisible by both 3 and 4, so it’s also divisible by 12.
Another example: Suppose we need to check if the number 224 is divisible by 12 or not
Check divisibility by 3: sum of digits of 224\(=2+2+4=8\), now 8 is not divisible by 3 (because 8 is not a multiple of 3) so, 224 is not divisible by 3.
Check divisibility by 4: number formed by last two digits of 224 is 24, now 24 is divisible by 4 (because 24 is a multiple of 4 as \(24÷4=6)\) so, 224 is also divisible by 4
Now we can see that the number 224 is not divisible by 3 but it is divisible by 4 only, so it is not divisible by 12.
Another example: Suppose we need to check if the number 549 is divisible by 12 or not
Check divisibility by 3: sum of digits of 549\(=5+4+9=18\), now 18 is divisible by 3 (because 18 is a multiple of 3 as \(18÷3=6)\) so, 549 is divisible by 3.
Check divisibility by 4: number formed by last two digits of 549 is 49, now 49 is not divisible by 4 (because 49 is not a multiple of 4) so, 548 is not divisible by 4
Now we can see that the number 548 is divisible by 3 but not divisible by 4, so it is not divisible by 12.
-- (Satinder Gill)