Divisibility Test

The divisibility test includes divisibility rules that are useful to check if a number is divisible by another number or not. It saves the time of performing long divisions to find multiples of a number. Such tests help to quickly identify the prime factorization. Divisors must completely divide the number without a remainder. Divisibility tests help to determine such divisions. There are various divisibility rules from 2 to 20.

Divisibility rule of 2

An even number or any number having an even number at one’s digit place will be divisible by 2. Examples: 8, 60, 276, 4682.

If a number 3684 is given, it can be represented as (3×1000) + (6×100) + (8×10) + 4×1.

Split it as a multiple of 10 + 4.

(368×10) + 4.

Since the last digit or 4 is a multiple of 2, the entire number is divisible by 2.

Divisibility rule of 3

If the sum of all the digits is a multiple of 3, the given number will be divisible by 3. Example: 354, 789, 2223.

In 2223, the sum of all digits = 2+2+2+3 = 9. Since 9 is divisible by 3, 2223 is also divisible by 3.

In 354, 354 = [(a multiple of 3) + 3] + [(a multiple of 3) + 5] + 4 = (a multiple of 3) + 3+5+4. Since, 3+4+5 = 12, a multiple of 3, so 354 is divisible by 3.

Divisibility rule of 4

If the last two digits of a number are multiples of 4 or 00, then the number will be divisible by 4. Example: 2644, 7836, 5900, 91572.

For 91572, we will consider the last two digits or 72. 72 is a multiple of 4 with a quotient of 18, 91572 is divisible by 4.

For 2644, the number is a multiple of 100 + the last two digits. So, (100×26) + 44 = multiple of 4 + 44. Since 44 is a multiple of 4 with a quotient, 2644 is divisible by 4.

Divisibility rule of 5

All the numbers with 5 or 0 at one’s digit place are divisible by 5. Example: 340, 295, 42670.

If a number 5765 is given, it can be represented as (5×1000) + (7×100) + (6×10) + 5×1.

Split it as a multiple of 10 + 4.

(576×10) + 5. The last digit is 5, so 5765 is divisible by 5.

Divisibility rule of 6

If a number is divisible by 2 and 3, it is also divisible by 6. Example: 468, 924, 7656.

For number 156, find the factors.

156 = 2×2×3×13

Since 2 and 3 are factors of 156, it is divisible by both 2 and 3. Thus, 156 is also divisible by 6.

Divisibility rule of 7

On subtracting the twice of the last digit with the remaining digits of the number, if the value is a multiple of 7, then the number will be divisible by 7. Example: 329, 1232, 7854.

For example, if the given number is 2457.

The last digit is 7, so twice of 7 is 14. 

245-14 = 231

Repeat the same. Twice of 1 is 2.

23-2 = 21

21 is a multiple of 7, so 2457 is divisible by 7.

Divisibility rule of 8

If the last three digits of a number are a multiple of 8 or 000, then the number will be divisible by 8. Example: 568, 3576, 9464.

For 78328, we will consider the last two digits or 328. 328 is a multiple of 8 with quotient 41, 78328 is divisible by 8.

For 3576, the number is a multiple of 1000 + the last 3 digits. So, (1000×3) + 576 = multiple of 8 + 576. Since 576 is a multiple of 8 with a quotient, 3576 is divisible by 8.

Divisibility rule of 9

If the sum of all the digits is a multiple of 9, the given number will be divisible by 9. Example: 351, 8568.

In 8568, the sum of all digits = 8+5+6+8 = 27. Since 27 is divisible by 9, 8568 is also divisible by 9.

Divisibility rule of 10

If the given number contains zero at one’s digit place, it is divisible by 10. Example: 340, 9670, 15870.

Divisibility rule of 11

Take the difference of the sum of alternate digits, and if it is a multiple of 11, then the number will be divisible by 11. Example: 121, 3179, 83853.

If the number is 678425,

Add digits at odd place, 6+8+2 = 16

Add digits at even place, 7+4+5 = 16

Difference of above-mentioned values = 16-16 = 00

Thus, 678425 is divisible by 11.

Divisibility rule of 12

If a number is divisible by both 3 and 4, it is also divisible by 12. Example: 744, 1056.

For example 1056 

The last two digits are divisible by 4. Sum of all the digits = 1+0+5+6 = 12. So it is divisible by 3. Hence, the number is divisible by 12.

Divisibility rule of 13

On multiplying the last digit with 4 and adding to the remaining digit, if the number is a multiple of 13, then it will be divisible by 13. Example: 143, 7293.

If the number is 5772, multiply last digit by 4 = 2×4= 8

Add 8 to the remaining digits, 577 + 8 = 585

Repeat the step – 5×4 = 20

58+20 = 78.

78 is a multiple of 13, 5772 is divisible by 13.

Divisibility rule of 17

On multiplying the last digit with 5 and subtracting it to the remaining digit, if the number is multiple of 17, then it will be divisible by 17. Example: 102, 1513.

If the number is 14484, multiply last digit by 5 = 4×5 = 20

Subtract 20 from remaining digits, 1448 – 20 = 1428

Repeat the step – 8×5 = 40

142-40 = 102

102 is a multiple of 17, so 14484 is divisible by 17.

Divisibility rule of 19

On multiplying the last digit with 2 and adding to the remaining digit, if the number is multiple of 19, then it will be divisible by 19. Example: 152, 6384.

If the number is 16188, multiply last digit by 2 = 8×2= 16

Add 16 to remaining digits, 1618+ 16 = 1634

Repeat the step – 4×2 = 8

163+8 = 171.

1×2 = 2

17+2 = 19

Thus, 16188 is divisible by 19.

Conclusion

These divisibility rules are helpful to find the factors quickly and thus making the calculations easy. With the help of divisibility tests, you do not have to perform long divisions.