Factors and Multiples
Imagine you have 12 chocolates and you want to share them equally among your friends. Can you share them equally among 2 friends? Yes, each gets 6. Among 3 friends? Yes, each gets 4. Among 4 friends? Yes, each gets 3. Among 5 friends? No, because 12 cannot be divided equally into 5 groups. The numbers that divide 12 equally (1, 2, 3, 4, 6, and 12) are called factors of 12. And from the other direction, 12 is a multiple of each of these numbers. Factors and multiples are two sides of the same coin. Understanding them is like having a superpower for maths because they help you simplify fractions, find common denominators, solve division problems, and much more. In this chapter, we will learn what factors and multiples are, how to find them, and explore their interesting properties. This topic is part of the Playing with Numbers chapter in Grade 6 Maths, and it lays the foundation for understanding prime numbers, HCF, and LCM.
What is Factors and Multiples - Grade 6 Maths (Playing with Numbers)?
Factor: A factor of a number is a whole number that divides it exactly (with no remainder). For example, 3 is a factor of 12 because 12 / 3 = 4 with no remainder. We also say 12 is exactly divisible by 3.
Multiple: A multiple of a number is the result of multiplying that number by any whole number. For example, the multiples of 4 are 4, 8, 12, 16, 20, 24, ... (which are 4x1, 4x2, 4x3, 4x4, 4x5, 4x6, ...)
Here is the connection: If A is a factor of B, then B is a multiple of A.
For example, 3 is a factor of 15 (because 15 / 3 = 5), and 15 is a multiple of 3 (because 3 x 5 = 15).
Finding all factors of a number: To find all factors of a number, start dividing it by 1, 2, 3, 4, and so on. Every number that divides it exactly is a factor. You can stop when you reach a number whose square is greater than the given number, because factors come in pairs.
For example, factors of 12: 12/1=12, 12/2=6, 12/3=4, 12/4=3 (repeated). So factors of 12 are 1, 2, 3, 4, 6, 12.
Key facts about 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.
Key facts about multiples: Every number is a multiple of itself (because n x 1 = n). Every number is a multiple of 1 (because 1 x n = n). The smallest multiple of any number is the number itself. There is no largest multiple because you can keep multiplying forever.
Factors and Multiples Formula
Relationship between factors and multiples:
If a x b = c, then a and b are both factors of c, and c is a multiple of both a and b.
Example: 4 x 5 = 20. So 4 and 5 are factors of 20, and 20 is a multiple of both 4 and 5.
Number of factors: Every number has a finite (limited) number of factors. For example, 12 has exactly 6 factors: 1, 2, 3, 4, 6, 12.
Number of multiples: Every number (except 0) has infinitely many multiples. The multiples of 3 are 3, 6, 9, 12, 15, ... and they never end.
Factor pairs: Factors come in pairs that multiply to give the number.
For 24: 1 x 24, 2 x 12, 3 x 8, 4 x 6. So the factor pairs are (1, 24), (2, 12), (3, 8), (4, 6).
Perfect Square Factor Rule: If a number is a perfect square, one of its factor pairs has both numbers the same. For 36: 6 x 6 is a factor pair. This means 36 has an odd number of factors.
Quick check: To check if A is a factor of B, divide B by A. If the remainder is 0, then A is a factor of B.
Divisibility: When we say "B is divisible by A," it means A is a factor of B, which means B / A leaves no remainder.
Derivation and Proof
Let us build a complete understanding of factors by finding all factors of 36, step by step.
We need to find all numbers that divide 36 exactly.
Start with 1: 36 / 1 = 36. So 1 and 36 are both factors. Factor pair: (1, 36).
Try 2: 36 / 2 = 18. So 2 and 18 are factors. Factor pair: (2, 18).
Try 3: 36 / 3 = 12. So 3 and 12 are factors. Factor pair: (3, 12).
Try 4: 36 / 4 = 9. So 4 and 9 are factors. Factor pair: (4, 9).
Try 5: 36 / 5 = 7 remainder 1. So 5 is NOT a factor.
Try 6: 36 / 6 = 6. So 6 is a factor. Factor pair: (6, 6). Since both are the same, 6 appears only once.
Try 7: 7 x 7 = 49 > 36. We can stop here because any factor greater than 6 would pair with a factor less than 6, and we have already found all of those.
So the factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36. That is 9 factors (an odd number, because 36 is a perfect square).
Now let us think about multiples of 6:
6 x 1 = 6, 6 x 2 = 12, 6 x 3 = 18, 6 x 4 = 24, 6 x 5 = 30, 6 x 6 = 36, 6 x 7 = 42, ...
The list goes on forever. Notice that 36 appears in this list (6 x 6 = 36), which confirms that 36 is a multiple of 6, and 6 is a factor of 36.
This shows the beautiful connection: the factor question ("Does 6 divide 36?") and the multiple question ("Is 36 in the list of multiples of 6?") are just two ways of asking the same thing.
Types and Properties
Here are the types of problems you will encounter:
Type 1: Finding All Factors - Given a number, list all its factors. Use systematic division starting from 1 and going up until you start repeating pairs.
Type 2: Listing Multiples - Write the first n multiples of a given number. Simply multiply the number by 1, 2, 3, 4, and so on.
Type 3: Checking If One Number is a Factor of Another - Divide the larger number by the smaller. If the remainder is 0, it is a factor.
Type 4: Finding Common Factors - Find factors that two or more numbers share. For example, factors of 12 are {1, 2, 3, 4, 6, 12} and factors of 18 are {1, 2, 3, 6, 9, 18}. Common factors are {1, 2, 3, 6}.
Type 5: Finding Common Multiples - Find multiples that two or more numbers share. For example, multiples of 3 are {3, 6, 9, 12, 15, 18, ...} and multiples of 4 are {4, 8, 12, 16, 20, 24, ...}. Common multiples are {12, 24, 36, ...}.
Type 6: Word Problems - Problems like sharing objects equally, arranging items in rows and columns, or finding when two events coincide (like two buses meeting at a stop).
Type 7: True/False and Fill in the Blanks - Statements about properties of factors and multiples that you need to verify.
Solved Examples
Example 1: Example 1: Finding All Factors of 48
Problem: Find all factors of 48.
Solution:
48 / 1 = 48 (Factor pair: 1, 48)
48 / 2 = 24 (Factor pair: 2, 24)
48 / 3 = 16 (Factor pair: 3, 16)
48 / 4 = 12 (Factor pair: 4, 12)
48 / 5 = 9 remainder 3 (5 is not a factor)
48 / 6 = 8 (Factor pair: 6, 8)
48 / 7 = 6 remainder 6 (7 is not a factor)
7 x 7 = 49 > 48, so we stop.
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 (10 factors)
Example 2: Example 2: Listing First 8 Multiples
Problem: Write 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
First 8 multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56
Example 3: Example 3: Checking if a Number is a Factor
Problem: Is 8 a factor of 72? Is 9 a factor of 50?
Solution:
72 / 8 = 9 (no remainder). Yes, 8 is a factor of 72.
50 / 9 = 5 remainder 5. No, 9 is NOT a factor of 50.
Example 4: Example 4: Finding Common Factors
Problem: Find the common factors of 18 and 24.
Solution:
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Common factors (numbers in both lists): 1, 2, 3, 6
The greatest common factor is 6. (This is the HCF, which you will study in detail later.)
Example 5: Example 5: Finding Common Multiples
Problem: Find the first 3 common multiples of 4 and 6.
Solution:
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
Multiples of 6: 6, 12, 18, 24, 30, 36, ...
Common multiples: 12, 24, 36, ...
First 3 common multiples of 4 and 6: 12, 24, 36
The smallest common multiple is 12. (This is the LCM.)
Example 6: Example 6: Sharing Equally (Word Problem)
Problem: Riya has 60 stickers. She wants to share them equally among her friends such that each friend gets at least 1 sticker and there are no stickers left over. In how many ways can she share them?
Solution:
She needs to find all factors of 60, as each factor represents a possible number of friends (or stickers per friend).
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Each factor gives a valid way to share: 1 friend gets 60 each, 2 friends get 30 each, 3 friends get 20 each, and so on.
She can share them in 12 different ways (corresponding to the 12 factors of 60).
Example 7: Example 7: Number of Factors
Problem: Which has more factors: 24 or 36?
Solution:
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Count = 8 factors.
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Count = 9 factors.
36 has more factors (9 vs 8).
Note: 36 is a perfect square (6 x 6), and perfect squares always have an odd number of factors.
Example 8: Example 8: Factor and Multiple Relationship
Problem: Fill in the blanks: (a) 5 is a factor of 35, so 35 is a ___ of 5. (b) 72 is a multiple of 8, so 8 is a ___ of 72.
Solution:
(a) 35 is a multiple of 5 (because 5 x 7 = 35).
(b) 8 is a factor of 72 (because 72 / 8 = 9).
Remember: If A is a factor of B, then B is a multiple of A. They are two sides of the same relationship.
Example 9: Example 9: Finding a Number from Factor Clues
Problem: I am a number between 20 and 30. I am a multiple of both 4 and 7. What number am I?
Solution:
Multiples of 4 between 20 and 30: 20, 24, 28.
Multiples of 7 between 20 and 30: 21, 28.
Common to both lists: 28.
The number is 28.
Check: 28 / 4 = 7 (yes, multiple of 4). 28 / 7 = 4 (yes, multiple of 7). And 20 < 28 < 30.
Example 10: Example 10: Perfect Squares and Odd Number of Factors
Problem: Show that 25 has an odd number of factors.
Solution:
Factors of 25: 25/1=25, 25/5=5. Factor pairs: (1,25) and (5,5).
Factors: 1, 5, 25. Count = 3 (odd number).
25 = 5 x 5 is a perfect square. The factor pair (5, 5) contributes only one factor to the list, making the total count odd.
25 has 3 factors, which is indeed an odd number.
Real-World Applications
Factors and multiples have many real-life applications. Whenever you need to divide things equally, you use factors. If a teacher has 30 notebooks to distribute equally among groups, the possible group sizes are the factors of 30: 1, 2, 3, 5, 6, 10, 15, 30.
Multiples are used whenever things happen at regular intervals. If a bus comes every 15 minutes starting from 8:00 AM, the bus times are at 8:00, 8:15, 8:30, 8:45, etc. These times are based on multiples of 15.
In music, rhythm and beats are based on multiples. A 4/4 time signature means you count in multiples of 4. Musical patterns repeat at regular multiples.
In construction, tiling a floor requires knowing factors. If a room is 12 feet by 8 feet, you need tiles whose dimensions are common factors of 12 and 8 (like 1, 2, or 4 feet) to fit perfectly without cutting.
In sports, when two runners with different lap times run on a track, they meet at the starting point at intervals that are common multiples of their lap times.
Factors and multiples form the basis for fractions (simplifying by common factors), LCM (finding common denominators), and many topics in higher mathematics.
Key Points to Remember
- 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.
- If A is a factor of B, then B is a multiple of A.
- 1 is a factor of every number. Every number is a factor of itself.
- The number of factors is always finite. The number of multiples is infinite.
- Factors come in pairs that multiply to give the number.
- Perfect squares have an odd number of factors.
- The smallest factor of any number is 1. The largest factor is the number itself.
- The smallest multiple of any number is the number itself. There is no largest multiple.
- Common factors of two numbers are factors that both numbers share. Common multiples are multiples that both numbers share.
Practice Problems
- Find all factors of 56.
- Write the first 10 multiples of 9.
- Is 7 a factor of 91? Check by division.
- Find all common factors of 20 and 30.
- Find the first 4 common multiples of 3 and 5.
- A gardener has 84 plants. He wants to plant them in equal rows. List all possible numbers of rows he can make.
- I am a number between 40 and 50. I have exactly 4 factors. What number(s) could I be?
- Is every factor of a number also a factor of its multiples? Explain with an example.
Frequently Asked Questions
Q1. How are factors and multiples related?
They describe the same relationship from opposite directions. If 4 is a factor of 20 (because 20 / 4 = 5), then 20 is a multiple of 4 (because 4 x 5 = 20). Whenever you find a factor relationship, you automatically have a multiple relationship, and vice versa.
Q2. Does every number have at least two factors?
Every number greater than 1 has at least two factors: 1 and the number itself. The number 1 is special because it has only one factor (itself). Numbers with exactly two factors (1 and the number) are called prime numbers.
Q3. Can a factor be greater than the number?
No. A factor of a number can be at most equal to the number itself. The largest factor of any number is the number itself (because n / n = 1). No number larger than n can divide n exactly.
Q4. Is 0 a multiple of every number?
Yes, technically 0 is a multiple of every number because any number times 0 equals 0. For example, 5 x 0 = 0, so 0 is a multiple of 5. However, in most school problems, we focus on positive multiples (starting from the number itself).
Q5. How do I quickly find all factors of a number?
Start dividing the number by 1, 2, 3, 4, and so on. Each time you get an exact division, both the divisor and the quotient are factors. Stop when the divisor exceeds the quotient (or when the square of the divisor exceeds the number). This gives you all factor pairs efficiently.
Q6. Can a number be a factor and a multiple of the same number?
Yes. Every number is both a factor and a multiple of itself. For example, 12 is a factor of 12 (12/12=1) and 12 is a multiple of 12 (12x1=12). This is because n x 1 = n and n / n = 1 for any number n.
Q7. Why do perfect squares have an odd number of factors?
Factors normally come in pairs. For 12: (1,12), (2,6), (3,4) - that is 6 factors (even). But for 36: (1,36), (2,18), (3,12), (4,9), (6,6) - the pair (6,6) has both elements the same, so 6 is counted only once. This gives 9 factors (odd). This happens whenever a number is a perfect square.
Q8. What is the difference between common factors and HCF?
Common factors of two numbers are ALL the factors they share. HCF (Highest Common Factor) is the largest of these common factors. For example, common factors of 12 and 18 are {1, 2, 3, 6}, and the HCF is 6 (the highest one).
Q9. Is 1 a factor of every number?
Yes, 1 divides every number exactly. Any number divided by 1 gives the number itself with no remainder. So 1 is a factor of every number. Similarly, every number is a multiple of 1.
Q10. How are factors used in simplifying fractions?
To simplify a fraction, you divide both the numerator and denominator by their common factors. For example, 12/18: common factors of 12 and 18 include 6. Dividing both by 6 gives 2/3. This is the simplest form because 2 and 3 have no common factor other than 1.
