Formula To Solve Divisibility Questions

Formulas for Divisibility & Definitions:

    A divisibility rule is a shorthand method of determining whether a given number is divisible by a fixed divisor without carrying out the division, usually by examining its digits.
    One whole number is divisible by another if, after dividing, the remainder is zero.
    If the whole number is divisible by another number than the second number is factor of 1st number.

Divisibility Formulas

     When we set up a division problem in an equation using our division algorithm, and r = 0, we have the  following equation:

a = bq

When this is the case, we say that a is divisible by b. If this is a little too much technical jargon for you, don’t worry! It’s actually fairly simple. If a number b divides into a number a evenly, then we say that a is divisible by b.

For example, 8 is divisible by 2, because 8 / 2 = 4. However, 8 is not divisible by 3, because 8 / 3 = 2 with a remainder of 2. We see that we can check to see if a number, a, is divisible by another number, b, by simply performing the division and checking to see if divides into a evenly.

formulas to solve divisibility questions

Divisibility Formulas & Divisibility Rules

Divisibility rule for 1

  • Every number is divisible by 1.
    Example: 5 is divisible by 1

Divisibility rule for 2

  • Any even number or number whose last digit is an even number (0, 2, 4, 6, 8) is divisible by
    Example: 220 is divisible by 2.

Divisibility rule for 3

  • A number is divisible by 3 if the sum of its digits is divisible by 3.
    Example: 315 is divisible by 3.
    Here, 3 + 1 + 5 = 9

    9 is divisible by 3. It means 315 is also divisible by 3.

Divisibility rule for 4

  • A number is divisible by 4, if the number formed by the last two digits is divisible by 4.
    Example: Example: 7568 is divisible by 4
    Here, 68 is divisible by 4 (68÷4 = 17)

Therefore, 7568 is divisible by 4

Divisibility rule for 5

  • A number is exactly divisible by 5 if it has the digits 0 or 5 at one’s place.
    Example: 5900, 57895, 4400, 1010 are divisible by 5.

Divisibility rule for 6

  • A number is exactly divisible by 6 if that number is divisible by 2 and 3 both. It is because 2 and 3 are prime factors of 6.
    Example: 63894 is divisible by 6, the last digit is 4, so divisible by 2, and sum 6+3+8+9+4 = 30 is divisible by 3.

Divisibility rule for 7

  • Double the last digit and subtract it from the remaining leading truncated number to check if the result is divisible by 7 until no further division is possible
    Example: 1093 is divisible by 7

Remove 3 from the number and double it = 6

Remaining number is 109, now subtract 109 from 6 = 109 – 6 = 103.

Repeat the process, We have last digit as 3, double = 6

Remaining number is 10, now subtract 10 from 6 = 10 – 6 = 4.

As 4 is not divisible by 7, hence the number 1093 is not divisible by 7.

Divisibility rule for 8

  • If the last three digits of a number are divisible by 8, then the number is completely divisible by 8.
    Example: 215632 is divisible by 8, as last three digits 632 is divisible by 8.

Divisibility rule for 9

  • It is the same as of divisibility of 3. Sum of digits in the given number must be divisible by 9.
    Example: 312768 is divisible by 9, Sum of digits = 3+1+2+7+6+8 = 27 is divisible by 9.

Divisibility rule for 10

  • Any number whose last digit is 0, is divisible by 10.
    Example: 10, 60, 370, 1000, etc.

Divisibility rule for 11

  • If the difference of the sum of alternative digits of a number is divisible by 11, then that number is divisible by 11.
    Example: 737 is divisible by 11 as 7 + 7 = 14 and 14 – 3 = 11, 11 is divisible by 11.
    416042 is divisible by 11 as 4 + 6 + 4 = 14 and 1 + 0+ 2 = 3, 14 – 3 = 11, 11 is divisible by 11.

Divisibility rule for 12

  • A number is exactly divisible by 12 if that number is divisible by 3 and 4 both.
    Example: 108 is divisible by 12. Sum of digit = 1 + 8 = 9, 9 is divisible by 3. And last two
    digits 08 is divisible by 4. Therefore, 108 is divisible by 12.

Divisibility rule for 13

  • Multiply the last digit with 4 and add it to remaining number in a given number, the result must be divisible by 13.
    Example: 208 is divisible by 13, 20 + (4*8) = 20 + 32 = 52, 52 is divisible by 13.

Divisibility rule for 14

  • A number is exactly divisible by 14 if that number is divisible by 2 and 7 both. It is because 2 and 7 are prime factors of 14.
    Example: 1246 is divisible by 14, as the last digit is even, so divisible by 2.
    Now check for 7,

Remove 6 from the number and double it = 12

Remaining number is 124, now subtract 124 from 12 = 112.

Repeat the process, We have the last digit as 2, double = 4

The remaining number is, now subtract 11 from 4 = 7

e. As 7 is divisible by 7, hence the number 1246 is divisible by 7.

Divisibility rule for 15

  • If the number divisible by both 3 and 5, it is divisible by 15.
    Example: 23505 is divisible by 15.
    Check for 3: 2 + 3 + 5 + 0 +5 = 15, 15 is divisibly by 3.
    Check for 5: It has the 5 at one’s place, therefore, divisible by 5.

Divisibility rule for 16

  • The number formed by last four digits in the given number must be divisible by 16.
    Example: 152448 is divisible by 16 as last four digits (2448) are divisible by 16

Divisibility rule for 17

  • Multiply the last digit with 5 and subtract it from remaining number in a given number, the
    result must be divisible by 17.
    Example: 136 is divisible by 17. 13 – (5 * 6) = 13 – 30 = 17, 17 is divisible by 17.

Divisibility rule for 18

  • If the number is divisible by both 2 and 9, it is divisible by 18.
    Example: 92754 is divisible by 18.
    Check for 2: the last digit is even, therefore, it is divisible by 2.
    .
    Check for 9: 9 + 2 + 7 + 5 + 4 = 27, 27 is divisible by 9.

Divisibility rule for 19

  • Multiply the last digit with 2 and add it to remaining number in a given number, the result must be divisible by 19.
    Example: 285 is divisible by 19, 28 + (2 *5) = 28 + 10 = 38, 38 is divisible by 19.

Divisibility rule for 20

  • The number formed by last two digits in the given number must be divisible by 20.
    Example: 245680 is divisible by 20.