Math Lesson 1.5.4 - Prime Factorization

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Welcome to our Math lesson on Prime Factorization, this is the fourth lesson of our suite of math lessons covering the topic of Multiples, Factors, Prime Numbers and Prime Factorization including LCM and GCF, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Prime Factorization

Prime factorization means writing a number as a product of its prime factors. For example, in the scheme shown in the previous paragraph in which the number 120 was split into factors, we must consider only the prime ones, that is 2, 2, 2, 3 and 5. When making the Proof:, we obtain

When the number 120 is expressed as a product of prime factors we can write this in a shorter way as 23 × 3 × 5 instead of 2 × 2 × 2 × 3 × 5.

Example 3

Write the following numbers as a product of prime factors.

1. 54
2. 90
3. 200

Solution 3

1. We can write:
   54 = 2 × 27
   = 2 × 3 × 9
   = 2 × 3 × 3 × 3
   = 2 × 33
2. We can write:
   90 = 2 × 45
   = 2 × 3 × 15
   = 2 × 3 × 3 × 5
   = 2 × 32 × 5
3. We can write:
   200 = 2 × 100
   = 2 × 2 × 50
   = 2 × 2 × 2 × 25
   = 2 × 2 × 2 × 5 × 5
   = 23 × 52

Another method for finding the prime factors of a number is by dividing it by prime numbers. These divisors are taken from the smallest, i.e. from 2 and, when it is not possible to divide the number by 2 anymore, we divide it by 3, 5, 7, 11, and so on, i.e. with other prime numbers. With this approach it is better to write the original number in one column, where quotients of divisions with prime numbers are placed below the original number while the prime factors are placed in the other column. For example, if we want to write all prime factors of 72, we write

Hence, the number 72 is written in prime factors as 2 × 2 × 2 × 3 × 3 = 23 × 32.

The method used above is known as the tabular method. It is very suitable for finding the prime factors of big numbers but it has other applications in math as we will see in the next paragraph.

Example 4

Write the numbers 54 and 64 as products of prime factors.

Solution 4

Using the tabular method explained earlier, we can write

So, we have 54 = 2 × 3 × 3 × 3 = 2 × 33.

Likewise,

So, we have 64 = 2 × 2 × 2 × 2 × 2 × 2 = 26.

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