Divisor 19727

To determine if any number is divisible by 19727, apply the "Sully 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 1084 and subtract this from R.
3. Take that result and cross off its final digit (units). Take this new number and subtract 5918 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 19727. That is, your original number is divisible by 19727 if (and only if) Y is. Now that it's much smaller, with basic knowledge of your 19727-times tables, it should be easy to visually see if Y is divisible by 19727. If the Y is still much larger than 19727, the above process can be repeated until it does reduce to within small multiples of 19727.

Easy!