Divisibility by 7

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A very well know rule is to double the last digit and subtract it from the remaining leading truncated number.

For example, 889, 9 is the last digit, 88 is the remaining leading truncated number.

9 * 2 = 18, and 88 - 18 = 70, which is a multiple of 7.

Hence 889 is divisible by 7.

Verify the following:
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A three-digit number is divisible by seven if (and only if) the sum of twice its most significant digit plus three times its middle digit plus its least significant digit is divisible by seven.


We need to explore other methods to deal with much bigger numbers.

Let's test whether the number 6065534139 is divisible by 7.
The key to the test is the number 546231 (= 3 * 7 * 19 * 37 * 37)

Under this number we write the number 546231 repeatedly:

6     0     6     5     5     3     4     1     3     9
6     2     3     1     5     4     6     2     3     1
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36    0    18    5    25    12    24    2    9     9

Under this we write the products of the two numbers in each column:
36, 0, 18, 5, 25, 12, 24, 2, 9, 9

Add these products together
36 + 0 + 18 + 5 + 25 + 12 + 24 + 2 + 9 + 9 = 140

140 is divisible by 7, hence 6065534139 is divisible by 7.