Prime Factorization

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?