Separate the number to be divided into groups of two digits, starting from the end and going on to the beginning. If there is a single digit remaining, keep it as it is.
Example: 10101 = 1 01 01, 999999 = 99 99 99.
Starting from the beginning add the first group of digits to the next, adding it to the tens of the next group and subtracting it from the units.
Example: 10101= 1 01 01 = 1 + 0 tens and 1 - 1 units and 01 = 10 01. Then, 10 01 = 10 tens and -9 units = 100 - 9 = 91. If the end result is divisible by 7, then the whole number is divisible by 7. If the end result is divisible by 13, then the whole number is divisible by 13. Here, the end result is divisible by 7 and 13 so the whole number is divisible by 7 and 13.
More examples: 99 99 99 = 99 +9 tens + 9 - 99 units and 99 = 1080 - 90 and 99 = 990 99 = 9999 - 990 = 9009 = 909 - 90 = 819 = 80 + 19 - 8 = 91.
343 = 30 + 43 - 3 = 70 divisible by 7.
117649 (7^6) = 110 + 76 = 186 - 11 = 175 = 1750 + 49 = 1799 - 175 = 1624 = 160+24 - 16 = 184-16 = 168 divisible by 7.
4826809 (13^6) = 4 82 68 09 = 40 + 82 - 4 = 118 = 118 68 09 = 1248-118 = 1130 09 = 11300+9 - 1130 = 11309 - 1130 = 10179 = 1 01 79. 10+1 = 11 - 1 = 10, 100 + 79 = 179 - 10 = 169 divisible by 13. 169 = 10+69 = 79 -1 = 78 = 6*13. This method can also be used to find the remainder by any division by 7 or 13, which would be the remainder of such division of the end result.
Example: 10101 = 1 01 01, 999999 = 99 99 99.
Starting from the beginning add the first group of digits to the next, adding it to the tens of the next group and subtracting it from the units.
Example: 10101= 1 01 01 = 1 + 0 tens and 1 - 1 units and 01 = 10 01. Then, 10 01 = 10 tens and -9 units = 100 - 9 = 91. If the end result is divisible by 7, then the whole number is divisible by 7. If the end result is divisible by 13, then the whole number is divisible by 13. Here, the end result is divisible by 7 and 13 so the whole number is divisible by 7 and 13.
More examples: 99 99 99 = 99 +9 tens + 9 - 99 units and 99 = 1080 - 90 and 99 = 990 99 = 9999 - 990 = 9009 = 909 - 90 = 819 = 80 + 19 - 8 = 91.
343 = 30 + 43 - 3 = 70 divisible by 7.
117649 (7^6) = 110 + 76 = 186 - 11 = 175 = 1750 + 49 = 1799 - 175 = 1624 = 160+24 - 16 = 184-16 = 168 divisible by 7.
4826809 (13^6) = 4 82 68 09 = 40 + 82 - 4 = 118 = 118 68 09 = 1248-118 = 1130 09 = 11300+9 - 1130 = 11309 - 1130 = 10179 = 1 01 79. 10+1 = 11 - 1 = 10, 100 + 79 = 179 - 10 = 169 divisible by 13. 169 = 10+69 = 79 -1 = 78 = 6*13. This method can also be used to find the remainder by any division by 7 or 13, which would be the remainder of such division of the end result.
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