Another method for divisibility by 7

We separate the candidate number into groups of two digits as usual, starting from the end. We add the second group of digits from the end to the first group, then add twice the next group of digits to the second group and once to the first group, then add 3 times the next group of digits to the third group, 3 times to the second group and once to the first group, then add 4 times the next group of digits to the fourth group, 6 times to the third group, 4 times to the second group and once to the first group and so on. Confusing? The times we add the digits are the numbers in Pascal's triangle save the last one, namely, 1. Think how inspired and inventive this method is! We then add the groups of numbers. If the end result is divisible by 7 or 49 then the candidate number is also divisible by 7 or 49, else, we have the remainder of the division.

Example: 140 = 1 40 = 1 40 + 1 = 1 41. 1 + 41 = 42, so 140 is divisible by 7.

Another example: 16807 (7^5) = 1 68 07 =1 68 68 + 7 = 1 68 75 = 1 68 + 2 75 + 1 = 1 70 76 = 1 + 70 + 76 = 147 = 1 1 + 47 = 1 48 = 1 + 48 = 49, divisible by 49 (note also that 147 is divisible by 49).