Divisor 27697

To determine if any number is divisible by 27697, apply this test:

1. If your number ("X") has 10 digits or more, separate the last (smallest) 9 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 10 digits, L = 0 and therefore R = X.
2. Multiply L by 185 and subtract this from R.
3. Take that result and cross off its final digit (units). Take this new number and subtract 8309 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 27697. That is, your original number is divisible by 27697 if (and only if) Y is. Now that it's much smaller, with basic knowledge of your 27697-times tables, it should be easy to visually see if Y is divisible by 27697. If the Y is still much larger than 27697, the above process can be repeated until it does reduce to within small multiples of 27697.

Easy!