Tests for Divisibility of Numbers

We say that a number is divisible by a second number if on dividing the first number by the second, we get a remainder of 0.

Divisibility by 1

Every number is divisible by 1.

E.g. On dividing 14 by 1, we get the quotient as 14 and a remainder as 0.

Divisibility by 2

Any even number is divisible by 2.

E.g. On dividing 36 (an even number) by 2, we get a quotient of 18 and a remainder of 0.

But on dividing 47 (not an even number) by 2, we get a quotient of 23 and a remainder of 1.

Divisibility by 3

If the sum of the digits of a number is a multiple of 3, then the number is divisible by 3.

E.g. On dividing 243 by 3, we get a quotient of 81 and a remainder of 0 since 2+4+3 = 9, which is divisible by 3.

But 313 is not divisible by 3. It gives a quotient of 104 and a remainder of 1, since 3+1+3 = 7 is not divisible by 3.

Divisibility by 4

Any two digit number can be easily checked to see whether it is divisible by 4 or not. For numbers having more than 2-digits, the number is divisible by 4 if the number formed by its last two digits (the digits in the ones and the tens place) is divisible by 4.

E.g. On dividing 2364 by 4, we get a quotient of 591 and a remainder of 0. We find that the number formed by the last two digits i.e., 64 is divisible by 4.

But the number 2350 is not divisible by 4. In this case we find, the number formed by the last two digits, i.e., 50 is not divisible by 4.

Divisibility by 5

Any number ending with a 5 or 0 is divisible by 5.

E.g. 645 is divisible by 5 whereas 347 is not divisible by 5.

Divisibility by 6

Any number, divisible both by 2 and 3, is divisible by 6.

E.g. 642 is divisible by 6 because it is divisible both by 2 and 3. In this case, the quotient is 107 and the remainder is 0.

Divisibility by 7

In order to check the divisibility of a number by 7, we take its last digit (the digit in the one's place) and double it. Next we subtract this number from the number formed by the remaining digits. If the result is divisible by 7 (0 is also divisible by 7), then the original number is divisible by 7. If we do not know whether the result is divisible by 7 or not, we repeat the process.

E.g. Let us try to find whether 4536 is divisible by 7 or not. For that, we take the last digit, 6, and double it. So we get 12. Our number formed by the remaining digits is 453. Therefore, we perform the subtraction 453-12 and get the result as 441. But, we do not know whether 441 is divisible by 7 or not, so we repeat the process. We take 1 (last digit), double it to get 2, subtract it from 44 to get a result of 42. Since 42 is divisible by 7, we can say that the number 4536 is divisible by 7. On dividing 4536 by 7, we get a quotient of 648 and a remainder of 0.

Divisibility by 8

Any number with 4 or more digits is divisible by 8, if the number formed by the last three digits is divisible by 8.

e.g. 9216 is divisible by 8. We find that the number formed by the last 3 digits, i.e., 216, is divisible by 8.

Divisibility by 9

Any number is divisible by 9, if the number formed by the sum of the digits be divisible by 9.

e.g. 8694 is divisible by 9, since 8+6+9+4 = 27, is divisible by 9. On dividing 8694 by 9, we get a quotient of 966 and a remainder of 0.

Divisibility by 10

Any number ending with 0, is divisible by 10.

e.g. 4570 is divisible by 10 whereas 457 is not divisible by 10.

Divisibility by 11

In this case, we find the difference between the sum of the digits in odd places (starting from the right, i.e. the one's place) and the sum of the digits in even places (from the right) of the number. If the difference is either 0 or is divisible by 11, then the number is divisible by 11.

E.g. Let the number be 1432563. First of all, we separate and add the numbers of the odd and the even places, starting from the right.

3+5+3+1 = 12 (taking the digits in the odd places)

6+2+4 = 12 (taking the digits in the even places)

Since we get a 0 on subtracting the two results, we can say that 1432563 is divisible by 11. On dividing 1432563 by 11, we get 130233 as the quotient and 0 as the remainder.

Divisibility by 12

Any number divisible by 3 and 4 is divisible by 12.

E.g. Let 2448 be the number. Since 2+4+4+8 = 18, it is divisible by 3. Also the number formed by taking the last two digits from the right is 48, which is divisible by 4. Hence, the number 2448 is divisible by 12. We get a quotient of 204 and a remainder of 0 when we divide 2448 by 12.

Divisibility by 13

We take the digit in the one's place and multiply it by 9. Next, we form a number from the remaining digits Then, we subtract the product and the newly formed number. If the result is divisible by 13, then the original number will also be divisible by 13.

E.g. Let the number be 416. The digit in the one's place is 6. We multiply 6 by 9 to get a product of 54. The number formed from the remaining digits is 41. Since 54 > 41, therefore, we subtract 41 from 54 to get 13 as the answer. Since 13 is divisible by 13, we say that 416 is divisible by 13. We could have subtracted 54 from 41 as well to get an answer of -13. -13 is also divisible by 13. On dividing 416 by 13, we get 32 as the quotient and 0 as the remainder.

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