Explainer: Prime Factorization with Exponents

In this explainer, we will learn how to use different strategies to find prime factorization using exponents.

You should already understand the terms prime number, composite number, and factor.

Definition: Prime, Composite, and Factor

  • Factors of a number are the numbers we multiply together to make the original number. We often write factors in pairs.
    For example, 6×1=6 and 2×3=6, so 1, 2, 3, and 6 are factors of 6. One factor pair for 6 is 1 and 6 because their product is 6; another factor pair is 2 and 3.
  • A prime number is a whole number which has exactly two factors: 1 and itself.
    For example, 7 is prime because the only factors of 7 are 1 and 7.
  • A composite number is a whole number with more than 2 factors.
    For example, 6 is a composite number because it has 4 factors.

We can write any number as a product of some of its factors. A useful way to write a number is as a product of only prime numbers.

Definition: Prime Factorization

The prime factorization of a number is a product of prime numbers which is equal to the original number.

For example, 2×3×5 is the prime factorization of 30 because 2, 3, and 5 are all prime and their product is 30.

Example 1: Finding the Prime Factorization of Small Numbers

Which of the following is the prime factorization of 18?

  1. 18
  2. 2
  3. 2×9
  4. 2×3×3
  5. 3

Answer

To verify which product is the prime factorization of 18, we must check two things: that the product is equal to 18 and that the factors in the product are all prime numbers.

  • Since 18 is not prime, that is not the correct answer.
  • Since neither 2 nor 3 are equal to 18, they are not correct.
  • Since 9 is not a prime number, 2×9 is not a prime factorization.

We are left with 2×3×3. Both 2 and 3 are prime, and the product is equal to 18. Hence, this is the prime factorization of 18.

The prime factorization of a number can be useful when we need to find the highest common factors or lowest common multiples of sets of numbers, but we will not discuss how to do this here. Instead, we will focus on two methods to find the prime factorization: factor trees and dividing by primes. These methods take advantage of the fact that when you have written a number as a product of its factors, you can always exchange one of the factors for a product which that factor is equal to.

For example, we know that 40=4×10. But we can substitute 4 for 2×2 or substitute 10 for 2×5 in the equation and the product on the right-hand side will still be equal to 40. This gives us a way of writing numbers as a product of smaller and smaller factors. If we continue with our example for 40, we will get to 40=2×2×2×5, at which point we cannot break down any of the factors into smaller numbers because they are all prime.

The key to finding prime factors is the following result.

Result: Factors of a Number

If 𝑎 is a factor of 𝑏, and 𝑏 is a factor of 𝑐, then 𝑎 is a factor of 𝑐.

For example, 6 is a factor of 12, and both 2 and 3 are factors of 6, which means that 2 and 3 are also factors of 12.

This result tells us that we can work with factors of our original number (which are smaller and easier to deal with than the original number) when looking for prime factors.

Now we will see how this helps us to find the prime factorization of a number by drawing factor trees.

Start with the number 60. Find any factor pair of 60 and write these two numbers on the first level of the tree. For example, 60=6×10.

Next, pick any number at the end of a branch, and check whether it is prime. If it is, then check a different number; if it is not, then factor it again. For example, 6 is not prime, so factorize 6 into 2×3 and write these on the next level of the tree.

Notice that, at any stage, the product of the numbers at the ends of branches is equal to the starting number. For example, in the above step, the factors at the ends of the branches are 2, 3, and 10, and 2×3×10=60.

Continue to factor composite numbers at the ends of branches until you are left with only prime numbers.

Once you have only prime numbers, you have found the prime factorization of the number. Write the number as a product of these factors; we can use either the multiplication sign or a dot to represent multiplication so, 60=2×3×2×5or60=2325.

Often, we write the prime factorization with the factors in ascending order, so 60=2×2×3×5or60=2235 or by using exponents to group factors that are the same, 60=22×3×5or60=2235.

We summarize the method of drawing a tree diagram below.

How to Find the Prime Factorization Using a Factor Tree

Step 1: First find any factor pair for the number, and write these numbers in the first two branches.

Step 2: For any factor that is not prime, write it as a product of two of its factors.

Step 3: Continue until all the branches end in prime numbers.

Step 4: The prime factorization is the product of all the prime numbers at the ends of the branches of the tree diagram. It is good practice to write the factors in the prime factorization in ascending order and to use exponents to simplify the expression.

Note: You can start with any pair of factors and you will get the same set of prime factors and the same prime factorization. The only thing that will change is the order in which you find the factors.

Next, we will see how to use the method of dividing by primes. This method is sometimes called the ladder method.

To find the prime factorization of 60, start by writing 60 in the first step (if it helps, you can think of each step like upside-down division). Then, write any prime factor of 60 next to it and the quotient underneath in a new step. So, we can choose 2 as a prime factor of 60, and then the quotient to write in the next step is 60÷2=30.

In the second step, we have to consider factors of 30 (which are also factors of 60). For our prime factor, we could choose either 2, 3, or 5. If we choose 3, then the quotient to write in the third step is 30÷3=10. We continue these steps until we obtain a quotient of 1. This shows us that there are no more prime factors of the number.

Once we reach a quotient of 1, we have identified all the prime factors of 60; 60 is the product of the prime factors written on the left-hand side of each step. Hence, 60=2×3×5×2=22×3×5.

Let’s summarize the method of dividing by primes.

How to Find the Prime Factorization By Dividing by Primes

Step 1: Write the number in the top step.

Step 2: Write a prime factor on the left, and divide by this prime factor to get a quotient in the step below.

Step 3: Continue finding prime factors and dividing by them until you reach a quotient of 1.

Step 4: Write the number as the product of the primes on the left-hand side.

Note that it does not matter what prime factors you choose at each step; the answer will always be the same. Also, you can record the steps in the method in a vertical line instead of stepping down and to the right each time.

Notice that there is only one prime factorization of a number. It does not matter which method you choose or which factors you choose in each step, you will always obtain the same set of prime factors at the end.

Next, we will show that these methods can be used to find the prime factorizations of larger numbers.

Example 2: Finding the Prime Factorization of Large Numbers

Find the prime factorization of 792.

Answer

There are a number of methods to find prime factorizations.

Method 1: We can use the method of dividing by primes to find each prime factor one step at a time.

Start by finding a prime factor of 792. Since 792 is even, we know that 2 is a prime factor. So, 792=2×396 and we record this as follows.

We know that all the prime factors of 396 will also be prime factors of 792, so the next step is to find a prime factor of 396 (for example 2) and divide by it, continuing to find prime factors of the results until there are no prime factors left.

Depending on the prime factors that are chosen at each step, the end result may look like the following.

Once we have a result of 1, we know that there are no more prime factors and that the prime factorization of 792 is 2×2×2×3×3×11.

Depending on which notation you choose to represent multiplication, we can write this using exponents as 23×32×11or233211.

Method 2: We can alternatively use factor trees to factor the number into primes in multiple steps.

To do this, start with 792 at the top of the tree and write a factor pair on the first two branches.

Then, for each number at the end of a branch, factor it further if it is not prime.

The numbers at the ends of the branches when you are finished are the prime factors of 792: 2×3×2×2×3×11 or, equivalently, 23×32×11.

Finally, observe that if you know the prime factorization of a number you can find all factor pairs using the associative and commutative properties.

For example, we can use our prime factorization of 792 to find factors of 792.

We know that 792=2×2×2×3×3×11. If we rearrange these factors using the commutative property, then we know we can group them and multiply in any order by the associative property. Hence, 792=(2×2×2)×(3×3×11)=8×33or792=(2×11)×(2×2×3×3)=22×36.

Thus, 8, 22, 33, and 36 are all factors of 792, and any factor of 792 which is greater than one will be equal to the product of a number of these prime factors.

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