Divisor 23

To determine if any number is divisible by 23, apply this test:

1. If your number ("X") has 6 digits or more, separate the last (smallest) 5 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 6 digits, L = 0 and therefore R = X.
2. Multiply L by 4 and subtract this from R.
3. Take that result and cross off its final digit (units). Take this new number and add 7 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 23. That is, your original number is divisible by 23 if (and only if) Y is. Now that it's much smaller, with basic knowledge of your 23-times tables, it should be easy to visually see if Y is divisible by 23. If the Y is still much larger than 23, the above process can be repeated until it does reduce to within small multiples of 23.

Easy!