Editing Divisibility rule (section)

====Case where the alternating sum of digits is used====

This method works for divisors that are factors of 10 + 1 = 11.

Using 11 as an example, 11 divides 11&nbsp;=&nbsp;10&nbsp;+&nbsp;1. That means <math>10 \equiv -1 \pmod{11}</math>. For the higher powers of 10, they are congruent to 1 for even powers and congruent to −1 for odd powers:

: <math>10^n \equiv (-1)^n \equiv \begin{cases}
 1 & \text{if } n \text{ is even}, \\
 -1 & \text{if } n \text{ is odd}
\end{cases} \pmod{11}.</math>

Like the previous case, we can substitute powers of 10 with congruent values:

: <math>1000\cdot a + 100\cdot b + 10\cdot c + 1\cdot d \equiv (-1)a + (1)b + (-1)c + (1)d \pmod{11},</math>

which is also the difference between the sum of digits at odd positions and the sum of digits at even positions.