Prime Factorization (original) (raw)

Last Updated : 26 Jan, 2026

Prime factorization means breaking a number down into the prime numbers that multiply together to make it.

• A prime number is a number greater than 1 that can only be divided by 1 and itself (like 2, 3, 5, 7, 11…).
• Prime Factorization involves only the prime numbers, as every composite number can be written as the product of primes.

**More Examples of Prime Factorization:

**12 can be written as 2 × 6
6 can be further factorized as 2 × 3.
So 12 can be rewritten as 2 × 2 × 3
No more factorization possible as 2 and 3 cannot be divide further, so 2 × 2 × 3 is our prime factorization

**54 can written as 2 × 27.
27 can be further factorized as 3 × 9.
So we rewrite 54 as 2 × 3 × 9
9 can be further factorized as 3 × 3.
So we rewrite 54 as 2 × 3 × 3 × 3.
No more factorization possible, so 2 × 3 × 3 × 3 is our prime factorization

When a number is expressed as a product of its prime factors, it is said to be in its prime factorization form.Prime Factorization Example Table

Prime factorization of some of the common composite numbers are:

Prime Factorization Methods

Two common methods of Prime Factorization are:

• Division Method
• Factor Tree Method

Prime Factorization by Division Method

In this method, the number is successively divided by prime numbers until the quotient becomes 1****,** with each division identifying a prime factor.

**Steps to identify the prime factors of a number by the Division Method :

• **Step 1: Divide the number by the smallest prime number (i.e. 2) until we are able to divide the given number without leaving any remainder.
• **Step 2: Move on to the next prime number and repeat the division until the quotient becomes 1.
• **Step 3: The prime factors are the divisors used in the division process.

Examples of Prime Factorization by Division Method

**Example 1: Find the Prime Factorization of 60 using the Division Method.

**Example 2: Find the Prime Factorization of 210 using the Division Method.

**Example 3: Express 56 as the product of its Prime Factors.

Prime Factorization by Factor Tree Method

The Factor Tree Method involves breaking down a number into its prime factors by constructing a tree-like structure called a factor tree.

**Steps to identify the prime factors of a number by the Factor Tree Method:

• **Step 1: Identify two factors of the number that are not prime.
• **Step 2: Write these two factors as branches of the factor tree.
• **Step 3: Repeat steps 1 and 2 for each non-prime factor until all branches end with prime numbers.
• **Step 4: The prime factors are the numbers at the end of the branches.

Examples of Prime Factorization by Factor Tree Method

**Example 1: Find the factorization of 60 by the Factor Tree Method.

**Example 2: Make the Factor Tree of 210.

Prime Factorization of Numbers

Some examples of prime factorization are listed below:

Finding HCF and LCM by Prime Factorization

HCF and LCM can be easily calculated by the method of prime factorization:

Finding HCF

For the HCF, take the lowest power of each common prime factor from both numbers.

For Example:

• Common prime factors: 2 and 3
• For 2: min⁡(2, 4) = 2
• For 3: min⁡(1, 1) = 1

So, the HCF is:

HCF = 22 × 31 = 4 × 3 = 12

Finding LCM

For the LCM, take the highest power of each prime factor present in either number.

For Example:

• Prime Factors: 2, 3, and 5
• For 2: max⁡(2, 4) = 4
• For 3: max⁡(1, 1) = 1
• For 5: max⁡(1, 0) = 1

So, the LCM is:

LCM = 24 × 31 × 51 = 16 × 3 × 5 = 240

Applications of Prime Factorization

• **Finding HCF and LCM: Prime factorization helps determine the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of numbers, essential for simplifying fractions and finding common denominators.
• **Cryptography: It is crucial in public key cryptography, such as RSA, where the difficulty of factoring large composite numbers ensures secure communication.
• **Simplifying Fractions: By factoring numerators and denominators into prime factors, common factors can be canceled out, simplifying fractions effectively.
• **Divisibility Rules: Prime factorization aids in applying divisibility rules, quickly indicating whether one number is divisible by another.
• **Data Compression: Techniques based on prime factorization can optimize data storage and transmission in computer science, enhancing efficiency.
• **Network Security: Algorithms based on prime factorization enhance data security during network transfers, protecting sensitive information.

Related Articles:

• Interesting Facts about Prime Numbers
• Factorization of Algebraic Expression
• Composite Numbers
• Real Life Applications of Prime Factorization
• Prime Factorization Tips and Tricks

Prime Factorization Solved Examples

**Problem 1: What is the Prime Factorization of 80?

**Solution:

To find the prime factorization of 80, we can start by dividing it by the smallest prime number, which is 2.

• 80 divided by 2 equals 40.
• 40 divided by 2 equals 20.
• 20 divided by 2 equals 10.
• 10 divided by 2 equals 5.

Now, since 5 is a prime number, we can stop dividing. Therefore, the prime factorization of 80 is: 2 × 2 × 2 × 2 × 5.

**Problem 2: Prime factorization of 120.

**Solution:

Starting with the smallest prime number, which is 2.

• 120 divided by 2 equals 60.
• 60 divided by 2 equals 30.
• 30 divided by 2 equals 15.
• Now, since 15 is not divisible by 2, we move on to the next prime number (i.e, 3)
• 15 divided by 3 equals 5.

Now, since 5 is a prime number, we can stop dividing. Therefore, the prime factorization of 120 is: 2 × 2 × 2 × 3 × 5

**Problem 3: What is the Factor Tree of 56?

Prime Factorization Unsolved Examples

**Example 1. Find the prime factorization of 36.

**Example 2. Determine the prime factorization of 90.

**Example 3. What is the prime factorization of 48?

**Example 4. Find the prime factorization of 105.

**Example 5. What is the prime factorization of 84?

**Example 6. Determine the prime factorization of 100.

**Example 7. Find the prime factorization of 2310.

**Example 8. What is the prime factorization of 56?

**Example 9. Determine the prime factorization of 150.

**Example 10. What is the prime factorization of 1250?