Divisor 3697

To determine if any number is divisible by 3697, apply the "Philip Test":

1. If your number ("X") has 9 digits or more, separate the last (smallest) 8 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 9 digits, L = 0 and therefore R = X.
2. Multiply L by 153 and subtract this from R.
3. Take that result and cross off its final digit (units). Take this new number and subtract 1109 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 3697. That is, your original number is divisible by 3697 if (and only if) Y is. Now that it's much smaller, with basic knowledge of your 3697-times tables, it should be easy to visually see if Y is divisible by 3697. If the Y is still much larger than 3697, the above process can be repeated until it does reduce to within small multiples of 3697.

Easy!