How do you know when you have found all of the factors of a number?

One method of finding the factors of a number is to perform a divisibility test on it, using the results to find pairs of factors. If you use this method, you know you've found all of the factors when you start going back towards factors you found at the beginning.

Example:

Take 84. We know that 1 is a factor of every number, and 1 goes into it 84 times, so 1 and 84 are factors.

Since it is even, 84 is divisible by 2. 84/2 = 42, so 2 and 42 are factors.

To test for divisibility by 3, add the digits together: 8+4=12. Since 12 is divisible by 3, 84 is also divisible by 3. 84/3 = 28, so 3 and 28 are factors.

4 will divide into 8 and 4 will divide into 4, so 84 is divisible by 4. 84/4 = 21, so 4 and 21 are factors.

84 does not end in 5 or 0, so it is not divisible by 5.

Since it is divisible by both 2 and 3, 84 is divisible by 6. 84/6 = 14, so 6 and 14 are factors.

84 is divisible by 7; 84/7 = 12, so 7 and 12 are factors.

84 is not divisible by 8.

To check for divisibility by 9, look at the sum of the digits again. 8+4 = 12; 12 is not divisible by 9, so neither is 84.

Since the number does not end in 0, 84 is not divisible by 10.

84 is not divisible by 11.

We already know that 84 is divisible by 12 (84/7 = 12); since we've come across one we've seen before, we know we are finished.

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