Divisibility rules are must know tools for the GMAT Math section – they are super efficient and time saving

Here is a summary of the main divisibility rules:

**To be divisible by :**

**2 : The number needs to be even**

**3 : The sum of the digits of the number should be divisible by 3**

**4 : The last two digits should be divisible by 4**

**5 : The last digit is either 0 or 5**

**6 : (should be divisible by 2 and 3) The number should be even and the sum of its digits should be divisible by 3**

**7 : Subtract twice times the last digit from the remaining number and recursively do it to reach a small enough number. This number should be divisible by 7. Example below.**

**8 : The last three digits are divisible by 8**

**9 : The sum of the digits is divisible by 9**

**10 : The last digit must be 0**

**11 : The sum of the digits in even positions subtracted from the sum of the digits in odd positions must be divisible by 11**

**12 : If the number is divisible by both 3 and 4**

**13 : 4 times the last digit of the number plus the number obtained by removing the last digit done recursively. Example below.**

**17 : Subtract 5 times the last digit of the number from the remaining number done recursively. Example below.**

The above are the main divisibility rules that may be tested on the GMAT Quant section. Readers will notice that any non-prime number’s (e.g., 14) divisibility rule can be derived from its prime factorization. In other words, to be divisible with 14, the number has to satisfy divisibility tests of 7 and 2.

**Examples:**

**Is 3437 divisible by 7?**

Lets use the appropriate recursive algorithm to get to a small enough number

343 – 2 x 7 = 329

32 – 2 x 9 = 14

14 is divisible by 7 and hence 3437 is divisible by 7

**Is 1599 divisible by 13?**

Lets use the appropriate recursive algorithm to get to a small enough number

159 + 4 x 9 = 195

19 + 4 x 5 = 39

39 is divisible by 13 and hence 1599 is divisible by 13

**Is 15878 divisible by 17**

Lets use the appropriate recursive algorithm to get to a small enough number

1587 – 5 x 8 = 1547

154 – 5 x 7 = 119

This is not divisible by 17 and hence 15878 is not divisible by 17

Let’s do a GMAT Math question and test our knowledge of divisibility rules:

**How many numbers 34ab9 are divisible by 99 where a and b are whole numbers?**

**A. 1**

**B. 2**

**C. 0**

**D. 3**

**E. None of the above**

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**Solution : To be divisible by 99, the number would have to be divisible by 9 and 11**

**Divisibility by 9**

3 + 4 + a + b + 9 should be divisible by 9

7 + a + b is divisible by 9

a + b either 2 or 11

a = 1, b = 1

a = 5, b = 6

b = 6, a = 5

a = 9, b = 2

a = 8, b = 3

a = 3, b = 8

a = 4, b = 7

a = 7, b = 4

**Divisibility by 11**

(3 + a + 9) – (4 + b) is divisible by 11

8 + a – b is divisible by 11

a – b = -8 or 3

Checking against potential a, b values identified from earlier

Only a = 7 and b = 4 satisfy the above conditions

We go with **option (A)**