Divisibility

Formulas for Divisibility Questions  Formulas for Divisibility in Aptitude

A divisibility  is a rule for finding whether the number is divisible by another number or not. We can say that when number a is divided by another number b and remainder becomes zero . Hence the number a is divisible by b  . It is away to find factors of a large numbers.In this Page Formulas for Divisibility is given.

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 $\frac{8}{2}$ = 4. However, 8 is not divisible by 3, because of $\frac{8}{3}$ = 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 b divides into an evenly.

 

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: 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 6 from 109 = 109 – 6 = 103.

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

Remaining number is 10, now subtract 6 from 10 = 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 x 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

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 divisible 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 x 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 x 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.

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