👁 Preview — Study, Practice and Revise are open; mock tests and the rest of the syllabus unlock on subscription. Unlock all · ₹4,999
← Back to Number System
Study mode

Prime Factorization

Study Practice Revise 🔒 Test

Introduction to Prime Factorization

In the study of numbers, understanding how to break down a number into its basic building blocks is essential. These building blocks are called prime numbers. Prime factorization is the process of expressing a number as a product of prime numbers. This concept is fundamental in quantitative aptitude and competitive exams because it helps solve problems related to divisibility, greatest common divisor (GCD), least common multiple (LCM), and simplifying fractions.

In this chapter, you will learn what prime and composite numbers are, how to perform prime factorization using different methods, and how to apply prime factorization to solve various problems efficiently. By mastering these techniques, you will develop a strong foundation for tackling a wide range of numerical problems.

Prime Numbers and Composite Numbers

Before we dive into prime factorization, let's understand the two key types of numbers involved:

  • Prime Numbers: A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. Examples include 2, 3, 5, 7, 11, and so on.
  • Composite Numbers: A composite number is a natural number greater than 1 that has more than two positive divisors. In other words, it can be divided evenly by numbers other than 1 and itself. Examples include 4, 6, 8, 9, 12, etc.

Understanding the difference is crucial because prime factorization involves breaking down composite numbers into prime numbers.

Comparison of Prime and Composite Numbers
Prime Numbers Composite Numbers
2 4
3 6
5 8
7 9
11 10
13 12
17 14
19 15
23 16
29 18
Key Concept

Prime Numbers

Numbers greater than 1 with exactly two divisors: 1 and itself.

Prime Factorization Process

Prime factorization is the process of expressing a composite number as a product of its prime factors. There are two common methods to perform prime factorization:

  • Factor Tree Method
  • Division Method (Repeated Division)

Both methods aim to break down a number into prime numbers step-by-step.

Factor Tree Method

This method involves breaking a number into two factors and then breaking those factors further until all factors are prime numbers. The process can be visualized as a tree, where the original number is the root, and branches represent factors.

84 12 7 3 4 2 2

In this factor tree for 84, we start by splitting 84 into 12 and 7. Since 7 is prime, we stop there. Next, 12 is split into 3 and 4. 3 is prime, so we stop. Finally, 4 is split into 2 and 2, both prime numbers. Thus, the prime factors of 84 are 2, 2, 3, and 7.

Division Method (Repeated Division)

This method involves dividing the number by the smallest possible prime number repeatedly until the quotient becomes 1. It is especially useful for larger numbers where factor trees become complex.

1
->

Step 1

Start dividing the number by the smallest prime (2)

2
->

Step 2

If divisible, write down the prime factor and divide the quotient again by the same prime

3
->

Step 3

If not divisible, move to the next prime number (3, 5, 7, ...)

4

Step 4

Repeat until the quotient becomes 1

Worked Examples

Example 1: Prime Factorization of 84 Easy
Find the prime factors of 84 using the factor tree method.

Step 1: Start by splitting 84 into two factors. For example, 12 and 7.

Step 2: Check if these factors are prime. 7 is prime, so keep it.

Step 3: Factor 12 further into 3 and 4.

Step 4: 3 is prime, so keep it. Factor 4 into 2 and 2.

Step 5: All factors are now prime: 2, 2, 3, and 7.

Answer: \(84 = 2 \times 2 \times 3 \times 7 = 2^2 \times 3 \times 7\)

Example 2: Prime Factorization of 210 using Division Method Medium
Find the prime factors of 210 using the division method.

Step 1: Divide 210 by the smallest prime number 2.

210 / 2 = 105, so 2 is a prime factor.

Step 2: Divide 105 by 2. Not divisible, move to next prime 3.

105 / 3 = 35, so 3 is a prime factor.

Step 3: Divide 35 by 3. Not divisible, move to next prime 5.

35 / 5 = 7, so 5 is a prime factor.

Step 4: Divide 7 by 5. Not divisible, move to next prime 7.

7 / 7 = 1, so 7 is a prime factor.

Answer: \(210 = 2 \times 3 \times 5 \times 7\)

Example 3: Finding GCD of 48 and 180 Medium
Find the greatest common divisor (GCD) of 48 and 180 using prime factorization.

Step 1: Prime factorize 48.

48 / 2 = 24, 24 / 2 = 12, 12 / 2 = 6, 6 / 2 = 3, 3 / 3 = 1

So, \(48 = 2^4 \times 3\)

Step 2: Prime factorize 180.

180 / 2 = 90, 90 / 2 = 45, 45 / 3 = 15, 15 / 3 = 5, 5 / 5 = 1

So, \(180 = 2^2 \times 3^2 \times 5\)

Step 3: Identify common prime factors with minimum exponents.

Common primes: 2 and 3

Minimum exponent of 2: \(\min(4,2) = 2\)

Minimum exponent of 3: \(\min(1,2) = 1\)

Step 4: Calculate GCD.

\(\mathrm{GCD} = 2^2 \times 3 = 4 \times 3 = 12\)

Answer: The GCD of 48 and 180 is 12.

Example 4: Finding LCM of 12 and 15 Medium
Find the least common multiple (LCM) of 12 and 15 using prime factorization.

Step 1: Prime factorize 12.

12 = \(2^2 \times 3\)

Step 2: Prime factorize 15.

15 = \(3 \times 5\)

Step 3: Identify all prime factors with maximum exponents.

Prime factors: 2, 3, 5

Maximum exponent of 2: 2 (from 12)

Maximum exponent of 3: 1 (common in both)

Maximum exponent of 5: 1 (from 15)

Step 4: Calculate LCM.

\(\mathrm{LCM} = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60\)

Answer: The LCM of 12 and 15 is 60.

Example 5: Simplifying Fraction 210/462 Hard
Simplify the fraction \(\frac{210}{462}\) using prime factorization.

Step 1: Prime factorize numerator 210.

210 = \(2 \times 3 \times 5 \times 7\)

Step 2: Prime factorize denominator 462.

462 / 2 = 231, 231 / 3 = 77, 77 / 7 = 11, 11 / 11 = 1

So, \(462 = 2 \times 3 \times 7 \times 11\)

Step 3: Cancel common prime factors.

Common factors: 2, 3, 7

After cancellation, numerator: \(5\)

After cancellation, denominator: \(11\)

Answer: \(\frac{210}{462} = \frac{5}{11}\)

GCD and LCM using Prime Factors

\[\mathrm{GCD}(a,b) = \prod p_i^{\min(a_i,b_i)} \quad \quad \mathrm{LCM}(a,b) = \prod p_i^{\max(a_i,b_i)}\]

GCD is product of common primes with minimum exponents; LCM is product of all primes with maximum exponents.

a,b = Numbers
\(p_i\) = Prime factors
\(a_i,b_i\) = Exponents in prime factorization

Tips & Tricks

Tip: Use factor trees for visual clarity when starting prime factorization.

When to use: When breaking down numbers into prime factors for the first time or for smaller numbers.

Tip: Memorize prime numbers up to 50 to quickly identify prime factors.

When to use: To speed up factorization and avoid unnecessary trial divisions.

Tip: Use the division method for larger numbers to avoid complex factor trees.

When to use: When dealing with numbers above 100 or when factor trees become cumbersome.

Tip: For GCD, always take the minimum powers of common prime factors.

When to use: While calculating GCD from prime factorizations.

Tip: For LCM, take the maximum powers of all prime factors involved.

When to use: While calculating LCM from prime factorizations.

Common Mistakes to Avoid

❌ Confusing prime and composite numbers (e.g., thinking 1 is prime).
✓ Remember that prime numbers have exactly two distinct positive divisors: 1 and itself. The number 1 is neither prime nor composite.
Why: Students often mistake 1 or composite numbers as prime due to lack of clarity on definitions.
❌ Stopping factorization too early, leaving composite factors unbroken.
✓ Continue factorization until all factors are prime numbers.
Why: Missing complete prime factorization leads to incorrect answers in GCD, LCM, and simplification problems.
❌ Mixing up rules for GCD and LCM exponents (using max for GCD or min for LCM).
✓ Use minimum exponents for GCD and maximum exponents for LCM calculations.
Why: Confusing these rules causes wrong results in problems involving GCD and LCM.
❌ Not applying divisibility rules before starting factorization.
✓ Use divisibility rules (e.g., divisibility by 2, 3, 5) to quickly identify prime factors.
Why: Skipping this step makes factorization longer and increases the chance of errors.
Curated videos per subtopic
Top YouTube explainers, AI-ranked for your exam and language. Unlocks with subscription.
Unlock

Try Practice next.

Progress tracking is paywalled — subscribe to mark subtopics as understood and save your streak.

Go to practice →
Ask a doubt
Prime Factorization · 10 free messages
Ask me anything about this subtopic. You have 10 free messages this session — chat history isn't saved in preview.