Factors of 120: Definition, Prime Factors & Pairs

The concept of factors of 120 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Learning how to find factors of 120 quickly builds number sense, helps in LCM/HCF questions, and improves calculation speed for competitive exams and school assessments.

What Are Factors of 120?

A factor of 120 is a whole number that divides 120 exactly, leaving no remainder. In other words, if you multiply two whole numbers and get 120, both those numbers are factors. This is useful for understanding composite numbers, finding common factors, and solving arithmetic or algebraic problems involving divisibility.

Complete List: The factors of 120 are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120.

Key Formula for Factors of 120

Here’s the standard formula using prime factors: \( 120 = 2^3 × 3^1 × 5^1 \)
All factors can be generated by taking all possible products of powers of 2 (0 to 3), 3 (0 to 1), and 5 (0 to 1).

Prime Factorization of 120

Prime factors of 120 are the building blocks for the number. Breaking 120 into only prime number multipliers lets us see its structure clearly.

1. Divide 120 by 2: 120 ÷ 2 = 60
2. 60 ÷ 2 = 30
3. 30 ÷ 2 = 15
4. 15 ÷ 3 = 5
5. 5 ÷ 5 = 1

So, the prime factors of 120 are 2 × 2 × 2 × 3 × 5 or more compactly, 2³ × 3 × 5.

Factor Pairs of 120

A factor pair of 120 consists of two whole numbers whose product is 120. For MCQs and mental maths, pair listing helps avoid missing factors.

How to Find Factors of 120 (Stepwise Method)

1. Start with 1 and 120 (since 1 × 120 = 120)
2. Test every number from 2 up to the square root of 120 (about 10.95).
   
   For each number n, if 120 ÷ n is whole, then n and 120÷n are both factors.
3. List these as factor pairs to avoid duplication and ensure you do not miss any.

For practice, follow the same method with another number, like the factors of 60.

Properties and Types of Factors of 120

Even factors: 2, 4, 6, 8, 10, 12, 20, 24, 30, 40, 60, 120
Odd factors: 1, 3, 5, 15
Prime factors: 2, 3, 5
Composite factors: 4, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
Universal factors: 1 and 120 (the smallest and largest possible factors of 120).

Remember, all factors (except 1 and 120) are composite, since 120 is a composite number. For more on prime factors and composite numbers, check out those pages for definitions and examples.

Prime Factorization Tree of 120

Using a factor tree, you can visualize the prime decomposition:

1. 120 breaks into 2 × 60
2. 60 breaks into 2 × 30
3. 30 breaks into 2 × 15
4. 15 breaks into 3 × 5 (both prime)

So the full breakdown is 2 × 2 × 2 × 3 × 5. This tree method is especially useful in factorization and LCM/HCF questions.

Speed Trick or Vedic Shortcut

Here's a fast way to find factors of any number like 120: List 1 and the number, then keep checking consecutive numbers (2, 3, 4...) up to the square root, pairing each with its complement. For timed exams, write pairs vertically to avoid repeats!

Example: Check if 7 divides 120: 120 ÷ 7 = 17.14 (not a whole number, so 7 is not a factor). If the division gives a decimal, skip to the next. This trick works for all numbers.

Try These Yourself

• Write all factor pairs of 120 including negative pairs.
• Check if 30 and 24 are factors of 120.
• Find the common factors of 60 and 120. (Tip: Use the common factors tool.)
• Write the sum of all factors of 120.

Solved Examples

Example 1: Which pair of factors of 120 add up to 23?
1. List factor pairs: (1,120), (2,60), (3,40), (4,30), (5,24), (6,20), (8,15), (10,12)

2. Check which sum is 23: 8 + 15 = 23
Final Answer: (8,15) is the pair.

Example 2: What is the highest common factor (HCF) of 90 and 120?
1. Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

2. Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

3. Common: 1, 2, 3, 5, 6, 10, 15, 30

4. HCF = 30

Example 3: What are the prime factors of 120?
120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5

Frequent Errors and Misunderstandings

• Missing factor pairs by stopping at 10 (must check all up to square root of 120).
• Confusing factors and multiples—remember, factors divide 120, while multiples are products like 240, 360, etc.
• Ignoring 1 and 120 (always include smallest and largest for completeness).

Relation to Other Concepts

The idea of factors of 120 connects with LCM and HCF, multiplication tables, and the table of 20. Mastering factors supports algebra and number theory in higher classes, and also helps in daily logical reasoning.

Classroom Tip

To quickly find all factors of 120, list pairs systematically: Start with 1, then try 2, 3, 4 ..., checking for no remainder. Vedantu’s teachers use visual tables and short tricks in live classes to encourage stepwise, error-free thinking.

We explored factors of 120—from basics, listing, prime factorization, tricks, to connections with LCM and practice examples. Continue learning and practicing with Vedantu to build confidence in math and ace competitive exams!

Useful Internal Links

• Factors of 60
• Prime Factors
• LCM and HCF
• Factorization