Divisibility rules/Rule for 11 proof

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A number $N$ is divisible by 11 if the alternating sum of the digits is divisible by 11.

Proof

An understanding of basic modular arithmetic is necessary for this proof.

Let $N = a_ka_{k-1}\cdots a_1a_0$ where the $a_i$ are base-ten numbers. Then $N = 10^k a_k + 10^{k-1} a_{k-1} + \cdots + 10 a_1 + a_0.$

Note that $10\equiv -1\pmod{11}$ . Thus $10^k a_k\!\! +\!\!10^{k-1} a_{k-1}\! +\! \cdots + 10a_1 + a_0 \equiv (-1)^k a_k + (-1)^{k-1} a_{k-1} + \cdots -a_1 + a_0 \pmod {11}.$

This is the alternating sum of the digits of $N$ , 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: $100a+10b+c$
we then can transpose this into: $99a+a+11b-b+c$
and that equals: $(99a+11b)+(a-b+c)$
which equals $11(9a+b)+(a-b+c)$
Since the first addend, $11(9a+b)$ will always be divisible by 11, we just need to make sure that $(a-b+c)$ is divisible by 11.

You can use this for any number. Here it is again, with and even-numbered digit number:

$1000a+100b+10c+d$
$1001a-a+99b+b+11c-c+d$
$(1001a+99b+11c)-(a-b+c-d)$
$11(91a+9b+c)-(a-b+c-d)$
So you just need to check $(a-b+c-d)$ for divisibility with 11.

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Divisibility rules/Rule for 11 proof

From AoPSWiki

Proof

See also