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

The last digit is even: 0, 2, 4, 6, or 8. For example, 128, 146, 34

Divisibility by 3

The sum of the digits is divisible by 3. For example, 102, 192, 99

Divisibility by 4

The last two digits is divisible by 4 or is 00. For example, 576, 144

Divisibility by 5

The last digit is either 0 or 5. For example, 155, 3970, 145

Divisibility by 6

The sum of the digits is divisible by 3 and the number itself is divisible by 2. It means that the number is divisible by both 2 and 3. For example, 714, 509796, 1728

Divisibility by 7

Subtract 2 times the last digit from the rest.

Group the numbers in three from units 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

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

Divisibility by 9

The sum of the digits is divisible by 9. For example, 1287, 11583, 2304

Divisibility by 10

The last digit or unit digit is 0. For example, 100, 170, 10590

Divisibility by 11

Add the digits in blocks of two from right to left. For example, 627. 6 + 27 = 33

The difference between the sums of digits in the odd and even places taken from right to left is either zero or a multiple of 11. For example, 17259; Sum of digits in even places
= 7 + 5 = 12, Sum of digits in the odd places = 1 + 2 + 9 = 12; Hence 12 – 12 = 0.

Divisibility by 12

It is divisible by 3 and by 4 both. For example, 672, 8064

Divisibility by 13

Group the numbers in three from units 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) 5 (4) 6 (5) 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 (5) cannot be determined

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 (5) 0

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.