Divisibility rules/Rule for 11 proof
A number is divisible by 11 if the alternating sum of the digits is divisible by 11.
An understanding of basic modular arithmetic is necessary for this proof.
Let where the are base-ten numbers. Then
Note that . Thus
This is the alternating sum of the digits of , which is what we wanted.
Here is another way that doesn't require knowledge of modular arithmetic. Suppose we have a 3-digit number that is expressed in the form:
we then can transpose this into:
and that equals:
Since the first addend, will always be divisible by 11, we just need to make sure that is divisible by 11.
You can use this for any number. Here it is again, with an even-numbered digit number:
So you just need to check for divisibility with 11.