Why does divisibility rule for 13 work?

Answer 1

Depends which rule you are talking about.

You can make a divisibility rule by making a balanced formula or factorising.

Say you wanted to make a divisibility test for 13 for a 3-digit number.

Let's call that number abc.

In actual fact, the number is

100a+10b+c=13V

(V just represents the idea that 100a+10b+c= a number which is divisble by 13).

So if the right side of the equation (13V) is divisble by 13, so is the left side.

First we will use factorising to make a rule. We can split the numbers on the left:

100a+10b+c=13V will be the same as:

91a+9a+13b-3b+c+c=13V

We know that 91a and 13b are divisible by 13, so we can take them out of the equation.

9a-3b+c=13V

So to test a 3-digit number we can use this.

Take the number 117

We can put it through the equation and get:

9-3+7

=13

13 is divisible by 13, so 117 is too.

Bigger equations can be made for numbers with more digits.

Another method is to create numbers of a/b/c which are divisible by 13 on the other side.

A 3-digit number example is:

100a+10b+c=130V-91c

this is equal to:

100a+10b-90c=130V

Now we can divide the whole thing by ten since it won't make a difference:

10a+b-9c=13V

And that's our formula.

Let's take a different 3-digit number

546.

50+4-54=0

0 counts as being divisible by 13 so it works!