Divisor 7723

To determine if any number is divisible by 7723, apply the "Mark 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 399 and subtract this from R.
3. Take that result and cross off its final digit (units). Take this new number and add 2317 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 7723. That is, your original number is divisible by 7723 if (and only if) Y is. Now that it's much smaller, with basic knowledge of your 7723-times tables, it should be easy to visually see if Y is divisible by 7723. If the Y is still much larger than 7723, the above process can be repeated until it does reduce to within small multiples of 7723.

Easy!