Lizzie came up with a divisibility test for a certain number $m \neq 1$: Break a positive integer $n$ into two-digit chunks, starting from the ones place. (For example, the number $354764$ would break into the two-digit chunks $64, 47, 35.$)

Find the alternating sum of these two-digit numbers, by adding the first number, subtracting the second, adding the third, and so on. (In our example, this alternating sum would be $64-47+35=52.$)

Find $m,$ and show that this is indeed a divisibility test for $m$ (by showing that $n$ is divisible by $m$ if and only if the result of this process is divisible by $m$).