Mastering Tests of Divisibility: A Complete Guide
Divisibility by 2
A number is said to be divisible by 2, if its units digit is any of 0, 2, 4, 6, 8
Another way we can say that, a number is divisible by 2 if the last digit is even (0, 2, 4, 6, or 8).
Example: Ex. 58694 is divisible by 2, while 86945 is not divisible by 2.
Example: 24 is divisible by 2 because 24 divided by 2 equals 12.
Divisibility by 3
A number is said to be divisible by 3, if the sum of its digits is divisible by 3.
Example for 123 : For example, the number 123 is divisible by 3 because 1 + 2 + 3 = 6, which is divisible by 3.
Example for number 695421 : the number 695421 is divisible by 3 because 6 + 9 + 5 + 4 + 2 + 1 = 27
27 is divisible by 3, so 695421 is divisible by 3.
Example for number 948653 : 9 + 4 + 8 + 6 + 5 + 3 = 35
35 is divisible by 3, so 948653 is divisible by 3.
Divisibility by 4
A number is divisible by 4, if the number formed by the last two digits is divisible by 4
For example, the number 512 is divisible by 4 because 12 is divisible by 4.
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Similarly, the number 972 is divisible by 4 because 72 is divisible by 4.
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6879376 is divisible by 4 because 76 is divisible by 4.
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496138 is not divisible by 4 because 38 is not divisible by 4.
Divisibility by 5
A number is divisible by 5, if its units digit is either 0 or 5
For example: 10 is divisible by 5 because its units digit is 0.
Similarly, 35 is divisible by 5 because its units digit is 5.
Divisibility by 6
A number is divisible by 6, if it is divisible by both 2 and 3
Example: 18 :
Reason: 18 is divisible by both 2 (18/2 = 9) and 3 (18/3 = 6) so it is divisible by 6.
Divisibility by 8
A number is divisible by 8, if the number formed by the last three digits of the given number is divisible by 8
For example: if we have the number 2,824, the last three digits are 824. Since 824 is divisible by 8, then 2,824 is divisible by 8.
Divisibility by 9
A number is divisible by 9, if the sum of its digits is divisible by 9
For example: if the number is 459, we can calculate the sum of its digits to be 4 + 5 + 9 = 18. Since 18 is divisible by 9, 459 is divisible by 9.
Divisibility by 10
A number is divisible by 10, it its last digit is 0
Example:
20 is divisible by 10 because its last digit is 0.
120 is divisible by 10 because its last digit is 0.
Divisibility by 11
A number is divisible by 11, if the difference of the sum of its digits at odd places and sum of its digits at even places is either 0 or a number divisible by 11
Example: Consider the number 2468.
Calculation:
Sum of digits in odd places = 2 + 6 = 8
Sum of digits in even places = 4 + 8 = 12
Difference = 12 - 8 = 4
Since 4 is divisible by 11, the number 2468 is divisible by 11.
Test Yourself: Consider the number 29435417, and consider the number 57463822
Divisibility by 12
A number is divisible by 12, if it is divisible by both 3 and 4
Example: 36 is divisible by 12 because it is divisible by both 3 (36/3 = 12) and 4 (36/4 = 9).
Divisibility by 14
A number is divisible by 14, if it is divisible by 2 as well as 7
Example: 28 is divisible by 14 because it is divisible by both 2 and 7.
Divisibility by 15
A number is divisible by 15, if it is divisible by 3 as well as 5.
Example: 75 is divisible by 15 because it is divisible by both 3 and 5.
Reason: 75 is divisible by 3 (75/3 = 25) and it is divisible by 5 (75/5 = 15). Thus, it is divisible by 15.
Divisibility by 16
If the last 4 digits of a number form a number is divisible by 16, then the number is divisible by 16
Example: Consider the number 87536.
The last 4 digits of this number are 7536. 7536 is divisible by 16, so 87536 is divisible by 16.
Reason: According to the divisibility by 15 rule, if the last 4 digits of a number form a number that is divisible by 16, then the number is divisible by 16.
In the number 463776, the number formed by last 4 digits, namely 3776, is divisible by 16. 463776 is divisible by 16.
In the number 895684, the number formed by last 4 digits, namely 5684, is not divisible by 16. 895684 is not divisible by 16.
Divisibility By 18
A number is divisible by 18, if it is divisible by both 2 and 9.
Divisibility By 24
A given number is divisible by 24, if it is divisible by both 3 and 8.
Divisibility By 40
A given number is divisible by 40, if it is divisible by both 5 and 8.
Divisibility By 80
A given number is divisible by 80, if it is divisible by both 5 and 16.