# Formulas for Divisibility

## Formulas for Divisibility in Aptitude

In this page we have provided the definition and formulas for Divisibility along with the rules of divisibility of all the important digits  asked in the exam.

## Formulas of Divisibility

### 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.
• 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.

### Rules of Divisibility

 Divisibility rule of 2 Any number whose last digit is an even number (0, 2, 4, 6, 8) is divisible by 2 Divisibility rule of 3 A number is divisible by 3 if the sum of its digits is divisible by 3. Divisibility rule of 4 A number is divisible by 4, if the number formed by the last two digits is divisible by 4. Divisibility rule of 5 A number is exactly divisible by 5 if it has the digits 0 or 5 at one’s place. Divisibility rule of 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. Divisibility rule of 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 Divisibility rule of 8 If the last three digits of a number are divisible by 8, then the number is completely divisible by 8. Divisibility rule of 9 It is the same as of divisibility of 3. Sum of digits in the given number must be divisible by 9. Divisibility rule of 11 If the difference of the sum of alternative digits of a number is divisible by 11, then that number is divisible by 11. Divisibility rule of 12 A number is exactly divisible by 12 if that number is divisible by 3 and 4 both. Divisibility rule of 13 Multiply the last digit with 4 and add it to remaining number in a given number, the result must be divisible by 13. Divisibility rule of 15 If the number divisible by both 3 and 5, it is divisible by 15. Divisibility rule of 17 Multiply the last digit with 5 and subtract it from remaining number in a given number, the result must be divisible by 17 Divisibility rule of 19 Multiply the last digit with 2 and add it to remaining number in a given number, the result must be divisible by 19.

Question 1 :Calculate how many numbers between 1 and 100, including both are divisible by 9 or 4.

A. 35
B. 33
C. 34
D. 30

Explanation :
11 numbers are divisible between 1 and 100 by 9.
25 numbers are divisible by 4 between 1 and 100.
Therefore, total numbers = 36.
Also, there are numbers which are divisible by both 4 and 9 and are counted twice.
= 11 + 25 – 2 = 34

Question 2 : Which of the following is exactly divisible by 11?

A. 817425
B. 4832817
C. 817259
D. 5533935

Explanation :
A given number is divisible by 11 only if the difference between the sum of the digits in the odd places and the sum of the digits in the even places is divisible by 11.
5+3+9+5= 22; 5+3+3= 11 (22-11= 11)

Question 3 : What will be the sum of remainders when 684 will be divided by 3 , 7 and 5?

A. 10
B. 9
C. 11
D. 6

Explanation :
684 when divided by 3 leaves a remainder of 0.
684 when divided by 7 leaves a remainder of 5.
684 when divided by 5 leaves a remainder of 4.
Thus sum of remainders = 0 + 5 + 4 = 9.

Question 4 : Find the number from the given numbers which is not divisible by 7:

A. 84
B. 126
C. 89
D. 161

Explanation :
All three numbers when divided by 7 leaves a remainder of 0 except for the number 89 which leaves a remainder of 5.

Question 5 : Find the second-smallest 3 digit number divisible by 2 and 7

A. 114
B. 128
C. 142
D. 107

Explanation :
114 is the smallest 3 digit number divisible by 2 and 7.
142 is the third smallest 3 digit number divisible by 2 and 7.
107 is only divisible by 7.
128 is the second smallest 3 digit number divisible by 2 and 7.

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