Prime Factorization: Methods and Examples

What Is Prime Factorization?

Prime factorization is the method of expressing a composite number as a sequence of prime numbers multiplied together.

To understand this concept, you must first understand factors. A factor is a whole number that divides into another number exactly, leaving no remainder.

A prime number has exactly two factors: $1$ and itself. This rule separates prime vs composite numbers. The numbers $2$, $3$, $5$, $7$, and $11$ are prime because they cannot be divided evenly by any other whole number.

Prime factors are simply the prime numbers that multiply together to build a specific given number. You can think of prime factors as the atoms of mathematics because they cannot be broken down any further.

Every whole number greater than 1 has exactly one unique prime factorization.

When to Use It

Finding the prime components of a number is a highly useful skill in mathematics, especially when working with multiple numbers at once.

You will use prime factorization to find the Greatest Common Factor and the Least Common Multiple of two or more numbers. Comparing the prime factorizations makes it easy to identify their common factors.

It is also the most reliable method for simplifying large fractions. By breaking the numerator and denominator into their prime components, you can instantly see which shared primes can be cancelled out.

Step-by-Step Method

There are two standard methods for finding prime factors. Both methods will always give you the exact same final result.

The first approach is the factor tree method. You split the starting number into any two factor branches, and continue splitting any composite factors until only prime numbers remain at the ends of the branches.

The second approach is the repeated division method. You divide the starting number by its smallest possible prime factor, and continue dividing the new quotients by prime numbers until you reach a final quotient of $1$. Knowing your divisibility rules makes this method much faster.

Once you have found all the prime factors, group identical numbers together and write them using exponents. This simplified format is called exponent form. For example, $2 \times 2 \times 3$ is written as $2^2 \times 3$.

Visual Worked Examples

Example 1: Using the factor tree method

Question: What is the prime factorization of $60$?

Method:

1. Choose any two numbers that multiply to make $60$, such as $6$ and $10$.
2. Split $6$ into $2 \times 3$. Both numbers are prime, so these branches stop.
3. Split $10$ into $2 \times 5$. Both numbers are prime, so these branches stop.
4. Collect all the prime numbers located at the ends of the branches.

Answer: $60 = 2 \times 2 \times 3 \times 5$, which is written as $2^2 \times 3 \times 5$.

Check: Multiply the values to verify the result: $4 \times 3 \times 5 = 12 \times 5 = 60$.

Example 2: Using the repeated division method

Question: What is the prime factorization of $126$?

Method:

1. Divide $126$ by the smallest prime, which is $2$. The quotient is $63$.
2. Since $63$ is not divisible by $2$, move to the next smallest prime, which is $3$. Divide $63$ by $3$ to get $21$.
3. Divide $21$ by $3$ to get $7$.
4. The number $7$ is prime. Divide $7$ by $7$ to get $1$.
5. The divisors used on the outside of the division steps form the prime factors.

Answer: $126 = 2 \times 3 \times 3 \times 7$, which is written as $2 \times 3^2 \times 7$.

Check: Multiply the values to verify the result: $2 \times 9 \times 7 = 18 \times 7 = 126$.

Example 3: Identifying a counterexample

Question: Is $2 \times 4 \times 5$ the correct prime factorization of $40$?

Method:

1. Check if the product equals the target number: $2 \times 4 \times 5 = 40$. The total is correct.
2. Check if every factor is a strictly prime number.
3. The number $2$ is prime, and the number $5$ is prime.
4. The number $4$ is composite because it can be divided evenly by $2$.

Answer: No. The composite number $4$ must be broken down further into $2 \times 2$. The correct factorization is $2 \times 2 \times 2 \times 5$, which is written as $2^3 \times 5$.

Check: $2^3 \times 5 = 8 \times 5 = 40$.

How to Check the Answer

You can easily verify any prime factorization by multiplying the expanded factors back together. If the product matches your original starting number, and you confirm that every base number is strictly prime, your answer is completely correct.

Always evaluate any exponents first before multiplying. For example, if you want to check the factorization $3^2 \times 5$, first expand $3^2$ to $9$. Then, multiply $9 \times 5$ to confirm the final result is $45$.

Common Mistakes

Avoid these frequent errors when finding prime factors:

• Including 1 in the final answer: The number $1$ is neither prime nor composite because it has only one factor. It should never appear anywhere in a prime factorization.
• Stopping before all factors are prime: It is easy to leave a number like $9$ or $4$ at the end of a factor tree branch. Always check carefully that every final leaf is a true prime number.
• Confusing addition with multiplication: Prime factorization requires multiplying the factors, not adding them. Writing $2 + 3 + 5$ instead of $2 \times 3 \times 5$ is incorrect and changes the mathematical meaning.