Divisor 11657

To determine if any number is divisible by 11657, apply the "Aygox Test":

1. If your number ("X") has 8 digits or more, separate the last (smallest) 7 digits from the rest. This makes two smaller numbers, call them L^eft and R^ight (note: don't add in trailing zeros to L). If there are fewer than 8 digits, L = 0 and therefore R = X.
2. Multiply L by 1706 and subtract this from R.
3. Take that result and cross off its final digit (units). Take this new number and subtract 3497 times the digit you just crossed off. Call this final result "Y".
4. Y will be much smaller than X, but we have preserved divisibility by 11657. That is, your original number is divisible by 11657 if (and only if) Y is. Now that it's much smaller, with basic knowledge of your 11657-times tables, it should be easy to visually see if Y is divisible by 11657. If the Y is still much larger than 11657, the above process can be repeated until it does reduce to within small multiples of 11657.

Easy!