# What is the divisibility rule for 11, 12, and 13?

Apr 5, 2016

#### Explanation:

Divisibility Rule for $11$

Divide the alternate digits in two different groups. Take the sum of alternate digits separately and find the difference of the two numbers. If the difference is $0$ or is divisible $11$, the number is divisible by $11$.

Example: $86456293$ is divided into two groups $\left\{8 , 4 , 6 , 9\right\}$ and $\left\{6 , 5 , 2 , 3\right\}$. Sum of the groups is $27$ and $16$, whose difference is $11$ and the it is divisible by $11$, $86456293$ is divisible by $11$.

Divisibility Rule for $12$

If the number is divisible by both $3$ and $4$, the number is divisible by $12$. Divisibility rule of $3$ is tat sum of digits is divisible by $3$ and divisibility rule of $4$ is that last two digits are divisible by $4$.

Example: In $185176368$ sum of all the digits is $45$ and is divisible by $3$ and also last two digits $68$ are divisible by $4$. As such the number $185176368$ is divisible by $12$.

Divisibility Rule for $13$

Recall the divisibility rule of $7$, this works for $13$ too.

Starting from right mark off the digits in groups of threes (just as we do when we put commas in large numbers).

Now add up alternate group of numbers and find the difference between the two. If the difference is divisible by $13$, entire number is divisible by $13$.

For example $123448789113$, these are grouped as $123$, $448$, $789$ and $113$

and $123 + 789 = 912$ and $448 + 113 = 561$.

As difference between $912 - 561 = 351$

As $351$ is divisible by $13$, $123448789113$ is divisible by $13$