Divisor 31

To determine if any number is divisible by 31, 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 2 and add to R.
3. Take that result and cross off its final digit (units). Take this new number and subtract 3 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 31. That is, your original number is divisible by 31 if (and only if) Y is. Now that it's much smaller, with basic knowledge of your 31-times tables, it should be easy to visually see if Y is divisible by 31. If the Y is still much larger than 31, the above process can be repeated until it does reduce to within small multiples of 31.

Easy!