Divisibility Rules

A divisibility rule is a method to determine whether a given integer is divisible by a fixed divisor without performing the division. You can do this by examining the digits of the integer.

Divisibility by 2

A number is divisible by 2 if the unit's digit is zero or divisible by 2. It also means that the last digit is even - 0, 2, 4, 6, or 8. For example, 128, 146, 34, 102.

Divisibility by 3

A number is divisible by 3 if the sum of digits in the number is divisible by 3. For example, 102, 192, 99.

For example, the number 3792 is divisible by 3 since 3 + 7 + 9 + 2 = 21, which is divisible by 3.

Divisibility by 4

A number is divisible by 4 if the number formed by the last two digits (ten’s digit and unit’s digit) is divisible by 4 or are both zero. For example, 576, 144.

For example, the number 2616 is divisible by 4 since 16 is divisible by 4.

Divisibility by 5

A number is divisible by 5 if the unit’s digit in the number is 0 or 5. For example, 155, 3970, 145.

Divisibility by 6

A number is divisible by 6 if the number is even and sum of its digits is divisible by 3. It means that the number is divisible by both 2 and 3. For example, 714, 509796, 1728.

For example, the number 4518 is divisible by 6 since it is even and sum of its digits 4 + 5 + 1 + 8 = 18 is divisible by 3.

Divisibility by 7

The unit digit of the given number is doubled and then it is subtracted from the number obtained after omitting the unit digit. If the remainder is divisible by 7, then the given number is also divisible by 7.

For example, consider the number 448. On doubling the unit digit 8 of 448 we get 16. Then, 44 - 16 = 28. Since 28 is divisible by 7, 448 is divisible by 7.

Group the numbers in three from unit's digit. Add the odd groups and even groups separately. The difference of the odd and even groups should be 0 or divisible by 7. For example, 85437954. The groups are 85, 437, 954. Sum of odd groups = 954 + 85 = 1039; Sum of even groups = 437; Difference = 602, which is divisible by 7.

Divisibility by 8

A number is divisible by 8, if the number formed by the last 3 digits is divisible by 8. For example, the number 41784 is divisible by 8 as the number formed by last three digits, 784 is divisible by 8.

If the hundred's digit is even, examine the number formed by the last two digits. If the hundred's digit is odd, examine the number obtained by the last two digits plus 4. For example, 512, 4096, 1304.

Divisibility by 9

A number is divisible by 9 if the sum of its digits is divisible by 9. For example, 1287, 11583, 2304.

For example, the number 19044 is divisible by 9 as the sum of its digits 1 + 9 + 0 + 4 + 4 = 18 is divisible by 9.

Divisibility by 10

A number is divisible by 10, if it ends in zero. For example, 100, 170, 10590.

For example, the last digit of 580 is zero, therefore, 580 is divisible by 10.

Divisibility by 11

A number is divisible by 11, if the difference of the sum of the digits at odd places and sum of the digits at even places is either zero or divisible by 11.

For example, in the number 38797, the sum of the digits at odd places is 3 + 7 + 7 = 17 and the sum of the digits at even places is 8 + 9 = 17. The difference is 17 – 17 = 0, so the number is divisible by 11.

Divisibility by 12

A number is divisible by 12 if it is divisible by 3 and 4. For example, 672, 8064.

Divisibility by 13

Group the numbers in three from unit's digit. Add the odd groups and even groups separately. The difference of the odd and even groups should be 0 or divisible by 13.

For example, 35250799415. The groups are 035, 250, 799, 415. Sum of odd groups = 035 + 799 = 834; Sum of even groups = 250 + 415 = 665; Difference = 834 – 615 = 169 which is divisible by 13.


Examples

Example 1: What is the remainder when 2354789341 is divided by 11?

Odd place digit sum (O) = 1 + 3 + 8 + 4 + 3 = 19.

Even place digits sum (E) = 4 + 9 + 7 + 5 + 2 = 27.

Difference (D) = 19 - 27 = -8

Remainder = 11 - 8 = 3.

Example 2: If 567P55Q is divisible by 88; Find the value of P + Q.

  1. 11
  2. 12
  3. 6
  4. 10

The number is divisible by 8 means; the number formed by the last 3 digits should be divisible by 8 which are 55Q. Only Q = 2 satisfy this.

From the divisibility rule of 11, (2 + 5 + 7 + 5) – (5 + P + 6) is divisible by 11.

So 8 - P is divisible by 11.

if P = 8, then only it is possible. So P = 8 and Q = 2.

So, P + Q = 10.

Example 3: If the first 100 natural numbers are written side by side to form a big number and it is divided by 8. What will be the remainder?

  1. 1
  2. 2
  3. 4
  4. 7

The number is 1234.....9899100

According to the divisibility rule of 8, we will check only the last 3 digits.

If 100 is divided by 8, the remainder is 4.

Example 4: What will be the remainder when 4444........44 times is divided by 7?

  1. 1
  2. 2
  3. 5
  4. 6

If 4 is divided by 7, the remainder is 4.

If 44 is divided by 7, the remainder is 2.

If 444 is divided by 7, the remainder is 3.

By checking like this, we come to know that 444444 is exactly divisible by 7.

So if we take six 4’s, it is exactly divisible by 7. Similarly twelve 4’s is also exactly divisible by 4 and 42 4’s will be exactly divisible by 7.

So out of 44, the remaining two 4's will give a remainder of 2.