A quick test for divisibility by 7

Question

Is 2,401 divisible by 7? Is 3,457? Is 100,002?

Find a rule for testing divisibility by 7 that involves only the digits of the number, prove that it works, and apply it to those three numbers. Then adapt the same idea to produce a test for divisibility by 13.

Accepted Answer

The rule: remove the last digit, subtract twice that digit from the remaining number, and repeat until the number is small. The original is divisible by 7 exactly when the final result is. Examples. 2,401: remove 1, leaving 240, subtract 2: 238. Remove 8, leaving 23, subtract 16: 7. Divisible by 7 (2,401 = 7 to the 4). 3,457: 345 minus 14 = 331; 33 minus 2 = 31; not divisible. 100,002: 10,000 minus 4 = 9,996; 999 minus 12 = 987; 98 minus 14 = 84; 8 minus 8 = 0. Divisible (100,002 = 7 × 14,286). Why it works. Write the number as 10a + b, where b is the last digit. The rule replaces it with a m…

A quick test for divisibility by 7

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