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How can you tell whether a number is divisible by 7? Most everyone knows how to easily tell whether a number is divisible by 2, 3, 5, or 9. A few less know tricks for testing divisibility by 4, 6, 8, or 11. But not many people have ever seen a trick for testing divisibility by 7.
Here’s the trick. Remove the last digit from the number, double it, and subtract it from the first part of the number. Do this repeatedly until you get something you recognize as being divisible by 7 or not.
For example, start with 432. Split into 43 and 2. Subtract 4 from 43 to get 39. Since 39 isn’t divisible by 7, neither is 432.
For another example, start with 8631. Split into 863 and 1. Subtract 2 from 863 to get 861.
Now split 861 into Split into 86 and 1. Subtract 2 from 86. Maybe you recognize 84 as a multiple of 7. If not, double 4 and subtract from 8 to get 0, which is divisible by 7. Either way, we conclude that 8631 is divisible by 7.
Why does this work? Let b be the last digit of a number n and let a be the number we get when we split off b. That says n = 10a + b. Now n is divisible by 7 if and only if n – 21b is divisible by 7. But n – 21b = 10(a – 2b) and this is divisible by 7 if and only if a – 2b is divisible by 7.
What about the remainder when you divide a number by 7? Here’s where the rule for 7 differs from the more familiar divisibility rules. For example, a number is divisible by 3 if its digit sum is divisible by 3, and furthermore the remainder when a number is divided by 3 is the remainder when its digit sum is divided by 3. But the divisibility rule for 7 does not give the remainder when a number is divided by 7. For a simple example, the divisibility rule for 31 terminates in 1, but the remainder with 31 is divided by 7 is 3.
Why doesn’t the divisibility rule for 7 give the remainder? It is true that 10a + b and (10a + b) – 21b have the same remainder when divided by 7. But then we factored this into 10(a -2b). It’s true that 10(a – 2b) is divisible by 7 if and only if (a – 2b) is divisible by 7, but if neither is divisible by 7 then they will leave different remainders.