Checking if a number is a multiple of 7

I just read a couple of nice tricks for checking if a number is divisible by 7. There are standard divisibility tests for 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12, but checking a number is divisible by 7 is somewhat more difficult. But these two tricks make the task more manageable. The proofs for these tricks can be found in the given links.

Method #1, from http://www.arscalcula.com/mental_math_divisibility_tests.shtml: Add multiples of 7 to get a multiple of 10, and then lop off the 0.

Here’s how it goes: You want to see whether, say, 11352 is divisible by 7 . To do this, first you either add or subtract a mutiple of 7 until you get a number ending in 0 . So in the case of 11352 , I would add 28 to get 11380 .

Now whack off the last zero, and repeat! So 11380 goes to 1138 . From that I subtract 28 to get 1110 , which goes to 111 . To that I add 49 to get 160 , which goes to 16 .Finally: 16 is not divisible by 7 and thus (this is the statement of the test), neither is 11352.

Method #2, from http://www.arscalcula.com/mental_math_divisibility.shtml: Separate the number into two parts: the ones digit, and everything else but the ones digit. Multiply the ones digit by 5, and add to the the second number.

It’s hard to understand what this means without seeing an example. Let n=434. Then 5 \cdot 4+43=63 . Since 63 is divisible by 7 , so is 434.

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