What is an easy way to find all the factors of a number?

Answer 1

If you know the prime factorization of a number, you can find out the total number of factors.
Example: 210
21 x 31 x 51 x 71 = 210
Add one to the exponents and multiply them.
2 x 2 x 2 x 2 = 16, the total number of factors.
You know that 1, 2, 3, 5, and 7 are factors. You need three more to get halfway.
2 x 3 = 6
2 x 5 = 10
2 x 7 = 14
Divide all those numbers into 210.
1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210

Answer 2

There is no universally easy (or hard) method. Some people might find one method easy and another difficult while others may well find the second easier and the first difficult.

The word "easier", in the question, implies that you know of at least one way but do not consider that to be particularly easy. But you have given no indication of what that way might be. I am therefore unable to work out what method you do know and offer an alternative.

Answer 3

Use some rules, then just guess.

All even numbers have 2 as a factor.

All numbers ending in 0 or 5 have 5 as a factor.

Add all the digits, if that is divisible by 3 or 9 then the original number will also have 3 or 9 as a factor.

If you know a list of squares, then (in your head) compare to the number you have.

Answer 4

There are a few ways to go about factoring. You can decide what works best for you. I always find the prime factorization first. Let's look at a random number: 108

The prime factorization can be found by using a factor tree.

108

54,2

27,2,2

9,3,2,2

3,3,3,2,2

2^2 x 3^3 = 108

Half of the factors will be less than the square root, half greater. If the number is a perfect square, there will be an equal number of factors on either side of the square root. In this case, the square root is between 10 and 11.

Adding one to the exponents of the prime factorization and multiplying them will tell you how many factors there are. In this case, the exponents are 2 and 3. Add one to each. 3 x 4 = 12

108 has 12 factors. Six of them are 10 or less, six of them are 11 or greater. All we have to do is divide the numbers one through ten into 108. If the result (quotient) turns out to be an integer, you've found a factor pair. Knowing the rules of divisibility will make that even easier.

108 is divisible by...

1 because everything is.

2 because it's even.

3 because its digits add up to a multiple of 3.

4 because its last two digits are a multiple of 4.

6 because it's a multiple of 2 and 3.

9 because its digits add up to a multiple of 9.

That's six factors less than 10. Divide them into 108. That's the rest of them.

(108,1)(54,2)(36,3)(27,4)(18,6)(12,9)

1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108

Notice that all of those numbers, except for 1, can also be found in the prime factorization.

Answer 5

Try dividing it by small prime numbers.

- 2 is a factor of an even number.

- If the digits add to a multiple of 3, then the 3 is a factor.

- If the number ends in 0 or 5, then 5 is a factor.

- Try dividing by 7, 11, 13 etc. if none of these work.

Answer 6

by finding the half of the number and list the factors all the way to that number