Factor Pairs
When two numbers are multiplied together to give a product, those two numbers form a factor pair. Finding all factor pairs of a number helps you understand its structure and is useful in division, fractions, and later in algebra.
In Class 4, you will learn to find all factor pairs of numbers up to 100 and discover patterns like how perfect squares have an odd number of factors.
What is Factor Pairs - Class 4 Maths (Factors and Multiples)?
A factor pair of a number is a pair of whole numbers that multiply together to give that number.
For example, the factor pairs of 12 are: (1, 12), (2, 6), (3, 4) because 1 × 12 = 12, 2 × 6 = 12, and 3 × 4 = 12.
If a × b = N, then (a, b) is a factor pair of N
Factor Pairs Formula
How to find all factor pairs of a number N:
Step 1: Start with 1. Since 1 × N = N, (1, N) is always a factor pair.
Step 2: Try 2. If N ÷ 2 has no remainder, then (2, N ÷ 2) is a factor pair.
Step 3: Try 3, 4, 5, ... Continue until the first number in the pair becomes equal to or larger than the second number.
Stop when the smaller factor meets or passes the larger factor — you have found all pairs.
Solved Examples
Example 1: Example 1: Factor pairs of 12
Problem: Find all factor pairs of 12.
Solution:
Step 1: 1 × 12 = 12 → (1, 12)
Step 2: 2 × 6 = 12 → (2, 6)
Step 3: 3 × 4 = 12 → (3, 4)
Step 4: 4 × 3 = 12 → already counted (3, 4). Stop here.
Answer: Factor pairs of 12: (1, 12), (2, 6), (3, 4)
Example 2: Example 2: Factor pairs of 18
Problem: Find all factor pairs of 18.
Solution:
Step 1: 1 × 18 = 18 → (1, 18)
Step 2: 2 × 9 = 18 → (2, 9)
Step 3: 3 × 6 = 18 → (3, 6)
Step 4: 4 × ? → 18 ÷ 4 = 4.5 (not a whole number). Skip.
Step 5: 5 × ? → 18 ÷ 5 = 3.6. Skip. Since 5 > 4.2 (√18 ≈ 4.2), stop.
Answer: Factor pairs of 18: (1, 18), (2, 9), (3, 6)
Example 3: Example 3: Factor pairs of a prime number
Problem: Find the factor pairs of 13.
Solution:
Step 1: 1 × 13 = 13 → (1, 13)
Step 2: 2 × ? → 13 ÷ 2 = 6.5. Not whole.
Step 3: 3 × ? → 13 ÷ 3 = 4.3. Not whole. 3 < 4 but 4 × 3 > 13. Stop.
Answer: 13 is a prime number. It has only one factor pair: (1, 13).
Example 4: Example 4: Factor pairs of a perfect square
Problem: Find the factor pairs of 36.
Solution:
Step 1: 1 × 36 = 36 → (1, 36)
Step 2: 2 × 18 = 36 → (2, 18)
Step 3: 3 × 12 = 36 → (3, 12)
Step 4: 4 × 9 = 36 → (4, 9)
Step 5: 6 × 6 = 36 → (6, 6)
Answer: Factor pairs of 36: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6). Note: (6, 6) means 36 is a perfect square.
Example 5: Example 5: Word problem — arranging chairs
Problem: Priya needs to arrange 24 chairs in equal rows. What are the different ways she can do this?
Solution:
Step 1: Find factor pairs of 24: (1, 24), (2, 12), (3, 8), (4, 6)
Step 2: Each pair gives a row arrangement:
- 1 row of 24 chairs
- 2 rows of 12 chairs
- 3 rows of 8 chairs
- 4 rows of 6 chairs
Answer: Priya can arrange chairs in 4 different ways.
Example 6: Example 6: Factor pairs of 1
Problem: Find the factor pairs of 1.
Solution:
Step 1: 1 × 1 = 1 → (1, 1)
Step 2: No other whole number divides 1 exactly.
Answer: The only factor pair of 1 is (1, 1). The number 1 has exactly one factor.
Example 7: Example 7: Finding a number from its factor pairs
Problem: A number has factor pairs (1, 48), (2, 24), (3, 16), (4, 12), (6, 8). What is the number?
Solution:
Step 1: Pick any pair and multiply: 6 × 8 = 48
Step 2: Verify: 1 × 48 = 48, 2 × 24 = 48, 3 × 16 = 48, 4 × 12 = 48 ✓
Answer: The number is 48.
Example 8: Example 8: Listing all factors from factor pairs
Problem: Factor pairs of 30 are (1, 30), (2, 15), (3, 10), (5, 6). List all factors of 30.
Solution:
Step 1: Write both numbers from each pair: 1, 30, 2, 15, 3, 10, 5, 6
Step 2: Arrange in order: 1, 2, 3, 5, 6, 10, 15, 30
Answer: Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 (8 factors).
Example 9: Example 9: Tiles on a wall
Problem: Aman has 40 square tiles. In how many ways can he arrange them in a rectangular pattern?
Solution:
Step 1: Find factor pairs of 40: (1, 40), (2, 20), (4, 10), (5, 8)
Step 2: Each pair gives a rectangle: 1 × 40, 2 × 20, 4 × 10, 5 × 8
Answer: Aman can make 4 different rectangular arrangements.
Example 10: Example 10: Comparing factor pairs
Problem: Which has more factor pairs — 20 or 25?
Solution:
Step 1: Factor pairs of 20: (1, 20), (2, 10), (4, 5) → 3 pairs
Step 2: Factor pairs of 25: (1, 25), (5, 5) → 2 pairs
Answer: 20 has more factor pairs (3) than 25 (2).
Key Points to Remember
- A factor pair is two numbers that multiply to give the original number.
- Every number has at least one factor pair: (1, itself).
- Prime numbers have exactly one factor pair: (1, the number).
- Perfect squares have a factor pair where both numbers are the same, e.g., (6, 6) for 36.
- To find all factor pairs, start from 1 and test each number. Stop when the smaller factor meets or exceeds the larger.
- Factor pairs are useful for arranging objects in rows and columns.
- From factor pairs, you can list all individual factors of a number.
Practice Problems
- Find all factor pairs of 20.
- Find all factor pairs of 50.
- How many factor pairs does the prime number 17 have?
- Meera wants to arrange 36 flowers in equal rows. List all possible arrangements.
- Find the factor pairs of 100.
- A number has factor pairs (1, 42), (2, 21), (3, 14), (6, 7). How many factors does it have?
- Which number between 10 and 20 has the most factor pairs?
Frequently Asked Questions
Q1. What is a factor pair?
A factor pair of a number N consists of two whole numbers that multiply to give N. For example, (3, 8) is a factor pair of 24 because 3 × 8 = 24.
Q2. How do you find all factor pairs of a number?
Start dividing the number by 1, 2, 3, 4, and so on. Each time the division gives a whole number, you have found a factor pair. Stop when the divisor becomes larger than the quotient.
Q3. How many factor pairs does a prime number have?
A prime number has exactly one factor pair: (1, the number itself). For example, 7 has only the factor pair (1, 7).
Q4. What is special about perfect squares and factor pairs?
A perfect square has a factor pair where both numbers are the same. For example, 49 has the factor pair (7, 7). This means perfect squares have an odd number of total factors.
Q5. Is (4, 6) the same factor pair as (6, 4)?
Yes. Since multiplication is commutative (4 × 6 = 6 × 4), these count as the same factor pair. By convention, we write the smaller number first: (4, 6).
Q6. How are factor pairs useful?
Factor pairs help in arranging objects in rectangular arrays, dividing items into equal groups, simplifying fractions, and understanding the structure of numbers.
Q7. What is the difference between factors and factor pairs?
Factors are individual numbers that divide evenly into a given number. Factor pairs are groups of two factors that multiply to give the number. From factor pairs, you can list all individual factors.
Q8. Does every number have a factor pair?
Yes. Every whole number greater than 0 has at least one factor pair. The number 1 has the factor pair (1, 1), and every other number has at least (1, itself).
Related Topics
- Factors of a Number
- Prime and Composite Numbers
- Multiples of a Number
- Common Factors
- Common Multiples
- HCF (Highest Common Factor) Introduction
- LCM (Least Common Multiple) Introduction
- Sieve of Eratosthenes
- Even and Odd Number Properties
- Factors and Multiples Word Problems
- Prime Numbers up to 100
- Patterns in Multiples
