How to Decompose a Number into Its Prime Factors

Every number is the product of a unique set of prime factors, a group of prime numbers (including repeats) that, when multiplied together, equals that number. You can find those prime factors for a given number, by using a process called decomposition.

An easy way to decompose a number is to make a factorization tree. Here’s how:

  1. Find two numbers that multiply to equal the original number; write them as numbers that branch off the original one.

    Knowing the multiplication table can often help you here.

  2. If either number is prime, circle it and end that branch.

  3. Continue branching off non-prime numbers into two factors; whenever a branch reaches a prime number, circle it and close the branch.

    When every branch ends in a circled number, you’re finished — just gather up the circled numbers.

Sample question

  1. Decompose the number 48 into its prime factors.

    48 = 2 x 2 x 2 x 2 x 3. Begin making a factorization tree by finding two numbers that multiply to equal 48:

    image0.jpg

    Continue making branches of the tree by doing the same for 6 and 8:

    image1.jpg

    Circle the prime numbers and close those branches. At this point, the only open branch is 4. Break it down into 2 and 2:

    image2.jpg

    Every branch ends in a circled number, so you’re finished. The prime factors are 2, 2, 2, 2, and 3.

Practice questions

  1. Decompose 18 into its prime factors.

  2. Decompose 42 into its prime factors.

  3. Decompose 81 into its prime factors.

  4. Decompose 120 into its prime factors.

Following are the answers to the practice questions:

  1. 18 = 2 x 3 x 3. Here’s one possible factoring tree:

    image3.jpg
  2. 42 = 2 x 3 x 7. Here’s one possible factoring tree:

    image4.jpg
  3. 81 = 3 x 3 x 3 x 3. Here’s one possible factoring tree:

    image5.jpg
  4. 120 = 2 x 2 x 2 x 3 x 5. Here’s one possible factoring tree:

    image6.jpg
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