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Prime Factorization
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Learning how to find the prime factorization of a number is important when we start learning about fractions. This is especially important when we start reducing fractions. Here are some vocabulary words that will help you with this lesson:

  • Factor: A number that divides evenly into another number (Example: 3 is a factor of 12 because \(12 \div 3\) is a whole number).
  • Prime Number: Any number where the only factors are 1 and itself (Example: 11 is a prime number. There is no other number other than 1 and 11 you can divide it by that will give you a whole number).

The first video will explain more about primes and how to determine if a number is prime.

Prime Numbers

Video Source (05:52 mins) | Transcript

Once we know what prime numbers are, we learn that each number is made up of smaller prime numbers. Breaking a number into the primes that make it is called its prime factorization. Every number has a prime factorization. For prime numbers, their only factors are themselves and 1. This video will show how to find the prime factorization of a number and work through a couple of examples.

Prime Factorization

Video Source (06:44 mins) | Transcript

It’s helpful to memorize some of the prime numbers, which will make it easier to find prime factorization in future problems. Use this chart to memorize the primes up to 20. Be familiar with the prime numbers up to 100.
This image is a grid with 10 boxes across and 20 boxes down. In each box are the numbers from 1 to 200. It starts with the number 1 in the top left corner. The numbers are filled in in order going to the right. The number 10 is in the top right corner. This pattern continues until we reach number 200 in the bottom right corner. This creates columns where all the numbers ending in the same digit are in the same column. The prime numbers are highlighted. They are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 113, 127, 131, 137 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199.

Additional Resources

Practice Problems

Find the prime factorization of the following numbers:
  1. 21 (
    Solution
    x
    Solution: \(21 = 3 \times 7\)
    Details:
    The number 21 is at the top of the image. Under 21 are two arrows pointing down and outward, one pointing to the right and one to the left. The arrow pointing to the left points down to the number 3. The arrow pointing down and to the right points to the number 7. There is an multiplication sign between the numbers 3 and 7 indicating that 3 times 7 is 21.
    \(3 \times 7 = 21\); and 3 and 7 are prime factors. The prime factorization of the number 21 is 3 and 7.
    )
  2. 13 (
    Solution
    x
    Solution: \(13 = 1 \times 13\)
    Details:
    The number 13 is at the top of the image. Under 13 are two arrows pointing down and outward, one pointing to the right and one to the left. The arrow pointing to the left points down to the number 1. The arrow pointing down and to the right points to the number 13. There is an multiplication sign between the numbers 1 and 13 indicating that 1 times 13 is 13.
    The definition of a prime number is any number where the only factors are 1 and itself.
    The only factors of 13 are 1 and itself, 13. The prime factorization of 13 is just 13.
    )
  3. 30 (
    Video Solution
    x
    Solution: \(30 = 2 \times 3 \times 5\)
    Details:

    (Video Source | Transcript)
    )
  4. 12 (
    Solution
    x
    Solution: \(12 = 2 \times 2 \times 3\)
    Details:
    First, find the factors of 12. Here we show a few ways to factor 12. It can be factored as either \(4 \times 3\) or \(6 \times 2\)
    This shows the first step in the prime factorization of 12. It starts with 12 at the top. Under the 12 are two arrows pointing down and outward in opposite directions. The arrow pointing down and out to the left points to a number 4. The arrow pointing down and out to the right points to a number 3. There is an x between the 4 and 3 indicating that 4 times 3 is 12.
    This displays a second version of the factorization of the number 12. It also starts with 12 at the top. Below this 12 is another set of downward pointing arrows. The arrow pointing down and to the left points to the number 6 and the arrow pointing down and to the right points to the number 2. There is an x between the 6 and the 2 indicating that 6 times 2 is 12.
    The numbers 3 and 2 are both prime, but 4 and 6 are not.
    Next, find the factors of 4 or 6. Notice how the prime factorization of 12 is \(2 \times 2 \times 3\) whether the starting factors were \(4 \times 3\) of \(6 \times 2\). The prime factorization is the same.
    This image is a continuation of the first version for the factorization of 12. Again it starts with 12 at the top with arrows pointing down to 4 on the left and 3 on the right. Below the 4 is another set of downward arrows. These arrows point to a 2 on the left and a two on the right. There is an x between the two twos meaning 2 times 2 is 4. The 2, 2, and 3 are all in red indicating they are all prime numbers.
    This is a continuation of the second version of the factorization of 12. Again, below the 12 is a set of downward pointing arrows. The arrow on the left points to a 6 and the arrow on the right points to a 2. Below the 6 is another set of downward pointing arrows. The arrow pointing down and to the left points to a 3 and the arrow pointing down and to the right is pointing to a 2. There is an x between the 3 and 2 indicating that 3 times 2 is 6. The numbers 3, 2, and 2 are all red because they are prime.
    )
  5. 54 (
    Solution
    x
    Solution: \(54 = 2 \times 3 \times 3 \times 3\)
    Details: 54 is an even number so we know it is divisible by 2.
    \(2 \times 27 = 54\)
    This shows the first step in the prime factorization of 54. It starts with 54 at the top. Under the 54 are two arrows pointing down and outward in opposite directions. The arrow pointing down and out to the left points to a number 2. The arrow pointing down and out to the right points to a number 27. There is an x between the 2 and 27 indicating that 2 times 27 is 54.
    The number 2 is prime, but 27 is not, so we must find the factors of 27.
    27 is divisible by 3
    \(3 \times 9 = 27\)
    This image is a continuation of the first version for the factorization of 54. Again it starts with 54 at the top with arrows pointing down to a red 2 on the left and black 27 on the right. Below the 27 is another set of downward arrows. These arrows point to a red 3 on the left and a red 9 on the right. There is an x between the 3 and 9 meaning 3 times 9 is 27.
    The number 3 is a prime number, but 9 is not prime, so we must find the factors of 9.
    9 is also divisible by 3
    \(3 \times 3 = 9\)
    This image is the third of three in the factorization of 54. Again it starts with 54 at the top with arrows pointing down to a red 2 on the left and black 27 on the right. Below the 27 is another set of downward arrows. These arrows point to a red 3 on the left and a black 9 on the right. There is an x between the 3 and 9 meaning 3 times 9 is 27. Under the 9 is a red 3 on the left and a red 3 on the right, with an x in between meaning 3 time 3 is 9. The red numbers indicate prime numbers.
    The number 3 is a prime number so there are actually two more 3’s in our prime factorization.
    Now we go back and find all the prime numbers in the factorization of 54.
    This image is the same as the previous image, third of three in the factorization of 54. Again it starts with 54 at the top with arrows pointing down to a red 2 on the left and black 27 on the right. Below the 27 is another set of downward arrows. These arrows point to a red 3 on the left and a black 9 on the right. There is an x between the 3 and 9 meaning 3 times 9 is 27. Under the 9 is a red 3 on the left and a red 3 on the right, with an x in between meaning 3 time 3 is 9. The red numbers have a box around each one, indicating prime numbers.
    \(54 = 2 \times 3 \times 3 \times 3\)
    )
  6. 250 (
    Video Solution
    x
    Solution: \(250 = 2 \times 5 \times 5 \times 5\)
    Details:

    (Video Source | Transcript)
    )

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Introduction to Fractions