Math Help: Finding Factors

Once you've learned your multiplication and division facts, you're ready to learn about factors. You can use your knowledge of factors to identify prime numbers and find the prime factorization of numbers. You'll also use factors when you study algebra!

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A number's factors are all of the numbers that it can be evenly divided by. Every whole number has at least two factors: itself and one. Prime numbers are numbers that have only themselves and one as factors. Composite numbers have three or more factors: themselves, one and at least one other number. For example, seven is a prime number because it has only two factors (seven and one). Eight is a composite number because it has one, two, four and eight as factors.

Factors vs. Prime Factorization

Don't confuse a number's factors with its prime factorization. A number's prime factorization is a list of prime numbers that have the original number as their product. For example, the prime factorization of eight is two times two times two, which is very different from the list of factors for eight.

Finding Factors: Examples for Beginners

Factors of 12 and 13

The number 12 is divisible by one, two, three, four, six and 12, so these are its factors. Since it has more than two factors, it's a composite number. The number 13 only has two factors: 13 and one. This makes it a prime number.

Factors of 24 and 25

The number 24 has one, two, three, four, six, eight, 12 and 24 as its factors, because it is divisible by all of these numbers. The number 25 is divisible by one, five and 25, so they are its factors. Since both 24 and 25 have more than two factors, they are composite numbers.

Finding the Factors of Large Numbers

If you have to find the factors for a big number, start small. When you find a small number that divides evenly into the big number, look at the resulting answer. Whatever numbers that number is divisible by, the larger number is divisible by as well. You can use these divisibility rules to help you:

  1. If a number's last digit is even, it can be divided by two.
  2. If you can add a number's digits to get a result that's divisible by three, the number itself is divisible by three.
  3. If a number's last two digits can be evenly divided by four, the number itself is divisible by four.
  4. If the number's last digit is five or zero, it is divisible by five.
  5. If a number is divisible by two and three, it's also divisible by six.
  6. If you can double a number's last digit, subtract that result from the remaining part of the number, and get either zero or a number that's divisible by seven, then the original number is divisible by seven.
  7. If you can divide the number's last three digits by eight, then the whole number can be divided by eight.
  8. If you can add up all of a number's digits and get a result that is divisible by nine, then the number is divisible by nine.

To practice these divisibility rules, apply them to the number 144. Then, read through the example below to see if you found all of this number's factors.

Finding Factors of 144

To find the factors of 144, apply the divisibility rules. For example, since 144 ends in an even number, it must be divisible by two. If you divide 144 by two, you get 72, so both two and 72 are factors of 144.

If you continue applying each divisibility rule, you'll see that 144 is also divisible by three, four, six, eight and nine. If you divide 144 by each of these numbers, you'll find that 48, 36, 24, 18 and 16 are also factors of 144.

Next, note that 48, 36 and 24 are all divisible by 12, so 144 must be divisible by 12 as well. Dividing 144 by 12 equals 12 (144 ÷ 12 = 12), so no new factors are revealed. Don't forget that one and 144 are also factors of 144, since every whole number is divisible by itself and one.

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