Divisor 691

To determine if any number is divisible by 691, 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 123 and add to R.
3. Take that result and cross off its final digit (units). Take this new number and subtract 69 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 691. That is, your original number is divisible by 691 if (and only if) Y is. Now that it's much smaller, with basic knowledge of your 691-times tables, it should be easy to visually see if Y is divisible by 691. If the Y is still much larger than 691, the above process can be repeated until it does reduce to within small multiples of 691.

Easy!