How do you find the factors and prime factors of a number?

Answer 1

A prime number is a whole number greater than 1 that is divisible by only itself and 1. So, for example, 5 is a Prime number because it is only divisible by 1 and 5, but 6 is not a prime number, because it is divisible by 1, 2, 3 and 6.

One number is a factor of another if it can evenly divide into it. So, for example, 6 is a factor of 18, since 18/6 is 3.

The prime factors of a number are all of the factors of that number that are prime numbers. So, to find the prime factors of 28, first list all of the numbers that are factors of 28. The factors of 28 are 1, 2, 4, 7, 14, and 28. Then write those factors that are prime: 2 and 7.

The prime factorization of a number is a little different. The prime factorization of a number is a unique set of prime numbers that multiply together to get the given number. So, the prime factorization of 28 is 2x2x7, because 2x2x7=28.

To find the prime factorization of a number, start by writing the number as a product of any pair of non-trivial factors. (So, any factors other than 1 and the number.) Then look at each of these numbers. If the number is prime, then it will be in the prime factorization of the given number. If it is not prime, write it as the product of any of its non-trivial factors. Continue until all numbers are prime.

For example, let's find the prime factorization of 28. We can write 28 as the product of the factors 4 and 7. Seven is a prime number, so it will be in the prime factorization of 28. Four, however is not a prime number, so we will write it as 2x2. Since 2 is a prime number, we don't have to "break down" these 2's any more. So the prime factorization of 28 is 2x2x7, or 7x2x2, or 2x7x2--the order of the prime numbers doesn't matter, but we usually write them from smallest to largest.

Can you find the prime factorization of 24? The answer is 2x2x2x3.

You only need to check the for factors less than (or equal to) the square root of the number. Also, you only need to check for prime factors - once you've got those, you can get the rest easily (if you want to).

Example: 1524

Try 2:

1524 / 2 = 762

Try 2 again:

762 / 2 = 381

Try 2 again:

It's not a factor. Move on to the next prime.

Try 3:

381 / 3 = 127

Try 3 again:

Not a factor.

Try 5:

Not a factor.

Try 7:

Not a factor.

Try 11:

Not a factor.

The next prime is 13. But we don't need to try 13, because 13 > sqrt(127). This means that if 13 had been a factor (say 13x=127), then x would have been a smaller factor of 127, and we would have removed it by now. Similarly, you don't need to try 17, or 19, or any more prime numbers. 127 is prime.

Conclusion: The prime factorisation of 1524 is 1524 = 2 * 2 * 3 * 127.