⚽ Live Match: maths/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:

Numbers	Prime Factorization	Numbers	Prime Factorization
36	2² × 3²	40	2³ × 5
24	2³ × 3	50	2 × 5²
60	2² × 3 × 5	48	2⁴ × 3
18	2 × 3²	30	2 × 3 × 5
72	2³ × 3²	42	2 × 3 × 7
45	3² × 5	20	2² × 5

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.

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

Prime Factorization Example By Division Method

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

Prime Factorization using Division Method

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.

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

Example of prime factorization using a factor tree

Example 2: Make the Factor Tree of 210.

Prime Factorization By Factor Tree Method

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

➢Practice: Solved Examples

Prime Factorization