Divisor 27691

To determine if any number is divisible by 27691, apply this test:

1. If your number ("X") has 7 digits or more, separate the last (smallest) 6 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 7 digits, L = 0 and therefore R = X.
2. Multiply L by 3124 and add to R.
3. Take that result and cross off its final digit (units). Take this new number and subtract 2769 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 27691. That is, your original number is divisible by 27691 if (and only if) Y is. Now that it's much smaller, with basic knowledge of your 27691-times tables, it should be easy to visually see if Y is divisible by 27691. If the Y is still much larger than 27691, the above process can be repeated until it does reduce to within small multiples of 27691.

Easy!