How do you know you have listed all the factors for a number?

Answer 1

If you know the prime factorization of a number, you can predict how many factors it has. Add 1 to each exponent of each prime factor and multiply. The product will be the number of factors.

Example: The prime factorization of 30 is 2 x 3 x 5

Each of those factors has an implied exponent of 1. Add 1 to each and multiply.

2 x 2 x 2 = 8

30 has eight factors.

Half of them will be less than the square root, half will be greater.

The square root of 30 is between 5 and 6.

By the rules of divisibility, you know that 30 is divisible by 1 (because everything is), 2 (because it's even), 3 ( because its digits add up to a multiple of 3), and 5 (because it ends in 0)

Dividing those numbers into 30 gives you 30, 15, 10 and 6

Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

That's how you know you got them all.

Answer 2

Use a systematic approach. For fairly small numbers, you can try division by each number, up to the specified number. But for larger numbers, it is better to factor the number into prime factors. All the factors of the number are the different combinations of the number itself. For example, 28 is equal to 2 squared times 7; so you can use the "2" zero, one, or two times as a factor; you can also use the "7" either zero or one times as a factor; so in total, you have 8 factors (those include the trivial factors, i.e., 1, and the number itself).