Divisor 1699

To determine if any number is divisible by 1699, apply the "Element Test":

1. If your number ("X") has 5 digits or more, separate the last (smallest) 4 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 5 digits, L = 0 and therefore R = X.
2. Multiply L by 194 and subtract this from R.
3. Take that result and cross off its final digit (units). Take this new number and add 170 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 1699. That is, your original number is divisible by 1699 if (and only if) Y is. Now that it's much smaller, with basic knowledge of your 1699-times tables, it should be easy to visually see if Y is divisible by 1699. If the Y is still much larger than 1699, the above process can be repeated until it does reduce to within small multiples of 1699.

Easy!