Secret Number Trick # 1

Everybody knows how to spot whether a number is divisible by 2, and most people learn how to tell whether a number is divisible by 3, 4, 5, 6, 8, 9 or 10.

But what about 7? Is it so difficult to figure out whether a number is divisible by 7? NO!

Suppose you are given a number N. To check divisibility of N by 7:

  1. Take the last digit of N and double it.
  2. Subtract this value from the number formed by considering N without its last digit.
  3. Check if 7 divides this. If so, then 7 also divides N, otherwise it doesn’t. (Tiny technicality: Note that 7 divides 0, as 0*7=0.)

The way I memorize this is by chanting the mantra “double and subtract, double and subtract, double and subtract”. Make sure that you subtract from N without its last digit, instead of just N.

 

For example, take 8162. Is this divisible by 7?

  1. The last digit of 8162 is 2. Doubling this gives 4.
  2. Subtracting 4 from 816 gives 812.
  3. Is 812 divisible by 7? Run the method again:
    1. The last digit of 812 is 2. Doubling this gives 4.
    2. Subtracting 4 from 81 gives 77.
    3. 77 is divisible by 7.

We conclude that 8162 is divisible by 7.

 

For a proof and a generalization of this method to all odd prime numbers, check out this document.

 

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