GMAT Math : Divisibility rules

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

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)