Factors and Multiples (Grade 5)
Factors and multiples are two closely related concepts that form the foundation for divisibility, HCF, LCM, and fractions.
A factor of a number divides it exactly with no remainder. A multiple of a number is obtained by multiplying it by a whole number. For example, 4 is a factor of 20, and 20 is a multiple of 4.
In Class 5, students learn to find all factors of numbers up to 100, list multiples, identify factor pairs, and understand the connection between factors and multiples.
What is Factors and Multiples - Class 5 Maths (Factors and Multiples)?
Factor: A number that divides another number exactly (with remainder 0).
Example: Factors of 12 are 1, 2, 3, 4, 6, and 12 because each divides 12 exactly.
Multiple: The result of multiplying a number by any whole number (1, 2, 3, ...).
Example: Multiples of 5 are 5, 10, 15, 20, 25, 30, ...
Key relationship:
If A is a factor of B, then B is a multiple of A.
Example: 6 is a factor of 42, so 42 is a multiple of 6.
Factor pairs: A factor pair is two numbers that multiply to give the original number.
| Number | Factor Pairs | All Factors |
|---|---|---|
| 24 | (1, 24), (2, 12), (3, 8), (4, 6) | 1, 2, 3, 4, 6, 8, 12, 24 |
| 36 | (1, 36), (2, 18), (3, 12), (4, 9), (6, 6) | 1, 2, 3, 4, 6, 9, 12, 18, 36 |
Factors and Multiples (Grade 5) Formula
If a x b = n, then a and b are both factors of n, and n is a multiple of both a and b.
Finding factors systematically:
Start dividing from 1 upward. Stop when the factor pair starts repeating (i.e., when the quotient becomes smaller than the divisor).
Types and Properties
Properties of Factors:
- 1 is a factor of every number.
- Every number is a factor of itself.
- The smallest factor of any number is 1.
- The largest factor of any number is the number itself.
- Factors are finite (limited in count).
- A factor is always less than or equal to the number.
Properties of Multiples:
- Every number is a multiple of itself (since n x 1 = n).
- Every number is a multiple of 1.
- The smallest multiple of any number is the number itself.
- Multiples are infinite (they go on forever).
- A multiple is always greater than or equal to the number.
Divisibility rules (shortcuts for finding factors):
| Divisible by | Rule | Example |
|---|---|---|
| 2 | Last digit is 0, 2, 4, 6, or 8 | 348 (last digit 8) |
| 3 | Sum of digits is divisible by 3 | 624 (6+2+4=12, divisible by 3) |
| 4 | Last 2 digits form a number divisible by 4 | 516 (16 / 4 = 4) |
| 5 | Last digit is 0 or 5 | 735 (last digit 5) |
| 6 | Divisible by both 2 and 3 | 462 (even, and 4+6+2=12) |
| 9 | Sum of digits is divisible by 9 | 729 (7+2+9=18, divisible by 9) |
| 10 | Last digit is 0 | 450 (last digit 0) |
Solved Examples
Example 1: Example 1: Finding All Factors
Problem: Find all factors of 48.
Solution:
Step 1: Start dividing from 1:
- 48 / 1 = 48 → pair (1, 48)
- 48 / 2 = 24 → pair (2, 24)
- 48 / 3 = 16 → pair (3, 16)
- 48 / 4 = 12 → pair (4, 12)
- 48 / 5 → not exact
- 48 / 6 = 8 → pair (6, 8)
- 48 / 7 → not exact
- 7 > 6 (repeating), so stop.
Answer: Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Example 2: Example 2: Listing Multiples
Problem: List the first 8 multiples of 7.
Solution:
7 x 1 = 7, 7 x 2 = 14, 7 x 3 = 21, 7 x 4 = 28, 7 x 5 = 35, 7 x 6 = 42, 7 x 7 = 49, 7 x 8 = 56
Answer: First 8 multiples of 7 = 7, 14, 21, 28, 35, 42, 49, 56
Example 3: Example 3: Factor or Not?
Problem: Is 8 a factor of 72? Is 7 a factor of 72?
Solution:
72 / 8 = 9 (exact, no remainder) → Yes, 8 is a factor of 72.
72 / 7 = 10 remainder 2 (not exact) → No, 7 is not a factor of 72.
Example 4: Example 4: Common Factors
Problem: Find the common factors of 18 and 24.
Solution:
Step 1: Factors of 18 = 1, 2, 3, 6, 9, 18
Step 2: Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
Step 3: Common factors = numbers in both lists = 1, 2, 3, 6
Answer: Common factors of 18 and 24 = 1, 2, 3, 6
Example 5: Example 5: Common Multiples
Problem: Find the first 3 common multiples of 4 and 6.
Solution:
Step 1: Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
Step 2: Multiples of 6: 6, 12, 18, 24, 30, 36, ...
Step 3: Common multiples = 12, 24, 36, ...
Answer: First 3 common multiples = 12, 24, 36
Example 6: Example 6: Using Divisibility Rules
Problem: Is 846 divisible by 2, 3, and 6?
Solution:
By 2: Last digit is 6 (even) → Yes
By 3: Sum of digits = 8 + 4 + 6 = 18 (divisible by 3) → Yes
By 6: Divisible by both 2 and 3 → Yes
Answer: 846 is divisible by 2, 3, and 6.
Example 7: Example 7: Word Problem — Equal Groups
Problem: Kavi has 60 marbles. He wants to put them in bags with the same number of marbles in each bag. In how many ways can he do this?
Solution:
Each way corresponds to a factor of 60.
Factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
He can put 1 per bag (60 bags), 2 per bag (30 bags), 3 per bag (20 bags), etc.
Answer: He can do this in 12 ways (one for each factor of 60).
Example 8: Example 8: Word Problem — Common Multiples
Problem: Meera visits her grandmother every 6 days. Neha visits the same grandmother every 8 days. They both visited on 1st March. When will they visit together again?
Solution:
We need the first common multiple of 6 and 8.
Multiples of 6: 6, 12, 18, 24, 30, ...
Multiples of 8: 8, 16, 24, 32, ...
First common multiple = 24
Answer: They will visit together again after 24 days, on 25th March.
Example 9: Example 9: Perfect Numbers
Problem: A perfect number is a number whose factors (excluding itself) add up to the number. Show that 28 is a perfect number.
Solution:
Factors of 28 = 1, 2, 4, 7, 14, 28
Sum of factors excluding 28 = 1 + 2 + 4 + 7 + 14 = 28
Answer: Since the sum equals the number, 28 is a perfect number.
Example 10: Example 10: Finding a Number from Clues
Problem: I am a 2-digit number. I am a multiple of 8. I am a factor of 96. My digits add up to 6. What number am I?
Solution:
Step 1: Factors of 96 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
Step 2: 2-digit multiples of 8 that are factors of 96: 16, 24, 32, 48
Step 3: Check digit sum = 6: 16 (1+6=7), 24 (2+4=6) ✓, 32 (3+2=5), 48 (4+8=12)
Answer: The number is 24.
Real-World Applications
Where factors and multiples are used:
- Equal sharing: Factors help determine all the ways to divide items into equal groups.
- Scheduling: Common multiples help find when events happening at different intervals coincide (e.g., bus timings).
- Fractions: Finding common factors simplifies fractions. Finding common multiples helps add fractions.
- Patterns: Multiples create repeating patterns — tiles, beads, colour sequences.
- Divisibility: Quickly checking whether a large number is divisible by 2, 3, 4, 5, 6, 9, or 10.
Key Points to Remember
- A factor divides a number exactly. Factors are finite.
- A multiple is the product of a number and a whole number. Multiples are infinite.
- If A is a factor of B, then B is a multiple of A.
- Every number has at least two factors: 1 and itself (except 1, which has only one factor).
- Use divisibility rules to quickly check factors for 2, 3, 4, 5, 6, 9, and 10.
- Common factors are factors shared by two or more numbers.
- Common multiples are multiples shared by two or more numbers.
- Factor pairs always come in two numbers that multiply to give the original number.
Practice Problems
- Find all factors of 72.
- List the first 10 multiples of 9.
- Find all common factors of 36 and 48.
- Is 156 divisible by 2, 3, 4, and 6? Use divisibility rules.
- Rahul has 84 stickers. In how many ways can he divide them equally among his friends?
- Find the first 3 common multiples of 5 and 8.
- I am a 2-digit number, a multiple of 9, and my digits add up to 9. Find all possible values.
- A bell rings every 12 minutes and another every 15 minutes. Both ring at 9:00 AM. When will they ring together again?
Frequently Asked Questions
Q1. What is the difference between a factor and a multiple?
A factor divides a number exactly (3 is a factor of 15 because 15 / 3 = 5). A multiple is what you get when you multiply a number by a whole number (15 is a multiple of 3 because 3 x 5 = 15). Factors are smaller than or equal to the number; multiples are larger than or equal to it.
Q2. How many factors does a number have?
It varies. Some numbers have few factors (prime numbers have exactly 2: 1 and themselves). Others have many — 60 has 12 factors. To find the count, list all factor pairs systematically.
Q3. How many multiples does a number have?
Every number has infinitely many multiples because you can keep multiplying by larger and larger whole numbers. The multiples of 3 are 3, 6, 9, 12, ... and they never end.
Q4. What are factor pairs?
A factor pair is two numbers that multiply to give the original number. For 20, the factor pairs are (1, 20), (2, 10), and (4, 5). Each pair gives all factors when listed out: 1, 2, 4, 5, 10, 20.
Q5. What are divisibility rules?
Divisibility rules are shortcuts to check if a number is divisible by another without doing the full division. For example, a number is divisible by 3 if the sum of its digits is divisible by 3. 456 has digit sum 15, which is divisible by 3, so 456 is divisible by 3.
Q6. What is a common factor?
A common factor is a number that is a factor of two or more numbers. For example, factors of 12 are 1, 2, 3, 4, 6, 12, and factors of 18 are 1, 2, 3, 6, 9, 18. The common factors are 1, 2, 3, and 6.
Q7. What is a common multiple?
A common multiple is a number that is a multiple of two or more numbers. The common multiples of 3 and 4 include 12, 24, 36, 48, etc. The smallest common multiple is called the LCM.
Q8. Is 1 a factor of every number?
Yes. Every number divided by 1 gives the number itself with no remainder. So 1 is a universal factor. Similarly, every number is a factor of itself.
Q9. How do factors help in simplifying fractions?
To simplify a fraction, divide both the numerator and denominator by their common factor. For 12/18, the common factors include 6. Dividing both by 6 gives 2/3. The largest common factor (HCF) gives the simplest form in one step.
Q10. What is a perfect number?
A perfect number is a number whose factors (excluding itself) add up to the number. The first two perfect numbers are 6 (1+2+3=6) and 28 (1+2+4+7+14=28). Perfect numbers are rare — the next ones are 496 and 8,128.
