Fast mental math secrets: division by 9

In this article, I'll show you how to perform division by at lightning speed in your head. You will learn how to quickly tell if a number is a multiple of and how to divide it by . And above all, I will show you why it works.

How to quickly detect a multiple of 9?

It’s so easy to check if a number is a multiple of that you might think it’s magic.

A number is a multiple of if and only if the sum of its digits is a multiple of .

For instance, , , are all multiples of and you can easily check using the above rule:

for , we have and for , .

There is a little catch with the exemple because and one might not easily know if is a multiple of . But we can applies the same rule to and we get .

This is called the digit sum of a number. It is obtained by repeatedly summing the digits of a number until we get a result under The above rule can be rewritten as follows:

A number is a multiple of if and only if its digit sum is equal to .

You might wonder why this rule stands true. You are right because there is no magic in math.

Proof of divisibility by rule

In this section I’ll explain the proof of the above rule related to the divisibility by .

It’s really simple. Take for instance. It can be rewritten as follows:

So we have where is clearly a multiple of and is nothing but the sum of ’s digits. The right-hand side of the equation is a multiple of if and only if is a multiple of , which is the case here.

The proof can be easily generalised. if are the digits of a number , we can write:

Finally, we have:

We can clearly see that since is a multiple of , is a multiple of if and only if is a multiple of where are ’s digits. The proof can be generalised to any number with more than 3 digits but I will not detail it to avoid overwhelming the novice reader. Fill free to ask me in the comment if you want me to go through the generalised proof.

Now that we can tell if a number is a multiple of and know why the rule works, how do we perform the division by ?

An ultra-fast method to divide by

In this section, you will learn a very simple method to calculate the division by of a number really fast, way more faster than using a calculator. First let’s consider multiples of .

Fast mental division by for multiples of

For instance let us compute . is a multiple of because . To find the result of , we perform the following steps:

Divide the sum of digits by :
Multiply the digits of the number from right to left respectively by and add up the results (we can ignore the right most digit): ⇒
Add results from 1. and 2.:

So .

Let us take another exemple:

(we can ignore the right most digit because it is multiplied by )

A final example:

The proof of this method is directly derived from the precedent section. Recall that if are the digits of a number , we can write:

is nothing but the step 1. of the method, is the step 2.

This proof can be easily generalised to any number of digits.

What if is not multiple of ? you might ask. Let’s find out in the next section.

Fast mental division by for non multiples of

Well, in this case, just find the nearest multiple of to that is less than . Then use the method for multiples of with the number found.

The steps are as follow with as an illustration.

Calculate the digits sum of : ⇒
Find the nearest multiple of less than by subtracting the digits sum from :
Apply the method for multiples of to the result found in 2.

Finally .

is nothing but the digits sum of .

Fast mental calculation can be intimidating if you don’t know the right methods. In this article, we have seen how easy it is to verify if a number is a multiple of and how to perform mental division by at lightning speed using a simple method. We’ve also learn why this method stands true.

If you want to practice mental calculation while having fun, use this really fun free game I’ve developed.