StudyOSStudyOS CBSE Study Material, Class 5–12

Factors and Multiples

NCERT Class 5 · Mathematics Based on NCERT Class 5 Mathematics textbook · Free CBSE study kit

Everything you need to ace this chapter, in one place.
Notes · Mind Map · Quiz · Assertion-Reason · Case Studies · Question Bank · Past Papers

Get Full Study Kit →
Studying with friends? Send them this chapter.

Chapter Notes

Overview
This chapter introduces the concepts of factors and multiples, which are fundamental to understanding number relationships in mathematics. It explains how factors are numbers that divide another number completely, while multiples are the products of a number with other whole numbers. The big idea is that every number has a unique set of factors and an infinite set of multiples, and common factors and common multiples help us solve real-world problems like jumping patterns and scheduling. Students must walk away knowing how to find factors, multiples, common factors, and common multiples, and understand the difference between prime and composite numbers.
Key Concepts (10)
Factor
A factor of a number is a number that divides the given number completely, leaving no remainder.
e.g. 2 is a factor of 12 because 12 ÷ 2 = 6, exactly.
Multiple
A multiple of a number is the product of that number with any whole number (1, 2, 3, ...).
e.g. 12 is a multiple of 3 because 3 × 4 = 12.
Common Factor
A common factor of two or more numbers is a number that divides each of them completely.
e.g. 2 is a common factor of 24 and 36 because 24 ÷ 2 = 12 and 36 ÷ 2 = 18.
Common Multiple
A common multiple of two or more numbers is a number that is a multiple of each of them.
e.g. 12 is a common multiple of 3 and 4 because 12 is in the table of 3 (3×4) and also in the table of 4 (4×3).
Prime Number
A prime number is a number that has exactly two factors: 1 and the number itself.
e.g. 13 is a prime number because its only factors are 1 and 13.
Array
An array is an arrangement of objects in rows and columns that shows the factors of a number.
e.g. For 15, an array of 3 rows and 5 columns shows factors 3 and 5.
Divisible
A number is divisible by another if the division leaves no remainder.
e.g. 28 is divisible by 4 because 28 ÷ 4 = 7, remainder 0.
Product
The product is the result of multiplying two or more numbers together.
e.g. The product of 3 and 5 is 15.
Multiplier
A multiplier is the number by which another number is multiplied.
e.g. In 4 × 7 = 28, the multiplier is 4.
Consecutive Numbers
Consecutive numbers are numbers that follow each other in order, like 12 and 13.
e.g. 12 and 13 are consecutive numbers.
Formulas & Equations (3)
Factor × Factor = Multiple
Reads as: Factor times factor equals multiple
Variables: Factor: a number that divides another completely; Multiple: the product of factors
Use when: When explaining the relationship between factors and multiples in any question
⚠ Students often write 'Multiple × Multiple = Factor' which is wrong.
Number × 1 = Number (The number itself is always a multiple of itself)
Reads as: Any number multiplied by 1 equals the number itself
Variables: Number: any whole number
Use when: When listing multiples or understanding that every number is a multiple of itself
⚠ Students forget that the number itself is a multiple, e.g., they list multiples of 6 as 6, 12, 18... but sometimes skip 6.
1 × Number = Number (1 is always a factor of any number)
Reads as: One times any number equals the number
Variables: Number: any whole number
Use when: When listing factors or understanding that 1 is a factor of every number
⚠ Students sometimes forget to include 1 in the list of factors.
How It Works
Finding factors using arrays
  1. Step 1: Choose a number, e.g., 12.
  2. Step 2: Arrange that many objects in rows and columns of equal size.
  3. Step 3: Each arrangement (e.g., 3 rows × 4 columns) shows two factors (3 and 4).
  4. Step 4: List all possible arrangements to find all factors.
💡 The number itself and 1 are always factors of any number. For 12, factors are 1, 2, 3, 4, 6, 12.
Finding common multiples using number line or times tables
  1. Step 1: List the first few multiples of each number using times tables.
  2. Step 2: Identify the numbers that appear in both lists.
  3. Step 3: The smallest common multiple is the first common multiple.
  4. Step 4: Continue the pattern to find more common multiples.
💡 Common multiples are infinite; the smallest one is often asked in exams.
Finding common factors using jumps on a number line
  1. Step 1: Start at 0 on the number line.
  2. Step 2: Take jumps of a certain step size (e.g., 2 steps).
  3. Step 3: Check if you can land exactly on both given numbers (e.g., 24 and 36).
  4. Step 4: If yes, the step size is a common factor of those numbers.
💡 A common factor exactly divides each number — no remainder allowed.
Exam Traps — Don't Make These Mistakes
⚠ Confusing factors and multiples
✗ Students write '12 is a factor of 3' instead of '3 is a factor of 12'.
→ A factor is smaller than or equal to the number; a multiple is larger. 12 is a multiple of 3, not a factor.
✓ 3 is a factor of 12 because 12 ÷ 3 = 4 exactly.
⚠ Forgetting 1 and the number itself as factors
✗ Students list factors of 12 as 2, 3, 4, 6 — missing 1 and 12.
→ Every number has 1 and itself as factors. The chapter explicitly states 'The number itself and 1 are always factors of any number'.
✓ Factors of 12 are 1, 2, 3, 4, 6, 12.
⚠ Thinking 0 is a factor
✗ Students write '0 is a factor of 12'.
→ 0 cannot be a factor of any number because division by 0 is undefined. The chapter states '0 cannot be a factor of any number'.
✓ 0 is not a factor of any number.
⚠ Mistaking common factor for common multiple
✗ Students say '12 is a common factor of 3 and 4'.
→ 12 is larger than both 3 and 4, so it cannot divide them. It is a common multiple, not a common factor.
✓ 12 is a common multiple of 3 and 4 because 3×4=12 and 4×3=12.
⚠ Assuming all even numbers have only even factors
✗ Students write 'Factors of even numbers must be even' as true.
→ Even numbers can have odd factors. For example, 6 is even but has factor 3 (odd). The chapter shows this is false.
✓ Factors of even numbers can be odd, e.g., 3 is a factor of 6.
⚠ Thinking multiples of odd numbers cannot be even
✗ Students write 'Multiples of odd numbers cannot be even' as true.
→ Odd numbers multiplied by an even number give an even multiple. For example, 3 (odd) × 2 = 6 (even). The chapter states this is false.
✓ Multiples of odd numbers can be even, e.g., 6 is a multiple of 3.
Compare & Contrast
Size relative to the original number
Factor
Multiple
A factor is always less than or equal to the original number, while a multiple is always greater than or equal to the original number.
What they represent
Common Factor
Common Multiple
A common factor divides each number exactly (smaller than or equal to the numbers), while a common multiple is a number that is in the times table of each number (larger than or equal to the numbers).
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 short answer (2-3 marks) on listing factors/multiples, 1 long answer (4-5 marks) on common factors and common multiples with word problems
Most asked: Finding factors and multiples of given numbers · Identifying common factors and common multiples · Prime numbers and their identification · True/false statements about factors and multiples
Don't skip: Word problems involving common multiples (e.g., animals jumping together, hunting schedules); True/false statements about properties of factors and multiples
One-Line Revision (read aloud before the exam)
  1. A factor divides a number exactly, leaving no remainder.
  2. A multiple is the product of a number with any whole number.
  3. Every number has 1 and itself as factors.
  4. The number itself is always a multiple of itself.
  5. Common factors divide all given numbers exactly.
  6. Common multiples appear in the times tables of all given numbers.
  7. Prime numbers have exactly two factors: 1 and the number itself.
  8. 0 cannot be a factor of any number.
  9. Factors of odd numbers cannot be even.
  10. One of the common multiples of two consecutive numbers is their product.
  11. The only common factor of any two consecutive numbers is 1.
  12. Multiples of odd numbers can be even if multiplied by an even number.
  13. Factors of even numbers can be odd.
  14. 12 is the first common multiple of 3 and 4.
  15. Common multiples are infinite; common factors are finite.

MCQs — 10 Questions with Answers

Q1. Which of the following is a factor of 20?

  • A. 4 ✓
  • B. 6
  • C. 7
  • D. 9

Answer: A — 4 is a factor of 20 because 4 × 5 = 20, while 6, 7, and 9 do not divide 20 exactly.

Q2. Which number is a multiple of 5?

  • A. 17
  • B. 25 ✓
  • C. 32
  • D. 43

Answer: B — 25 is a multiple of 5 because 5 × 5 = 25, while the others cannot be made by multiplying 5 by a whole number.

Q3. What are the common factors of 12 and 18?

  • A. 1, 2, 3, 6 ✓
  • B. 1, 2, 4, 6
  • C. 2, 3, 6, 12
  • D. 1, 2, 3, 9

Answer: A — Factors of 12 are 1, 2, 3, 4, 6, 12 and factors of 18 are 1, 2, 3, 6, 9, 18; the common ones are 1, 2, 3, 6.

Q4. Which of the following is NOT a factor of 24?

  • A. 3
  • B. 5 ✓
  • C. 6
  • D. 8

Answer: B — 5 does not divide 24 exactly because 24 ÷ 5 = 4.8 with a remainder, while 3, 6, and 8 all divide 24 exactly.

Q5. Which number is a common multiple of 4 and 6?

  • A. 18
  • B. 20
  • C. 24 ✓
  • D. 26

Answer: C — 24 is a common multiple because 4 × 6 = 24 and 6 × 4 = 24, while 18 is only a multiple of 6, 20 is only a multiple of 4, and 26 is neither.

Q6. What are the common factors of 15 and 25? (Fill in the blanks: _____ and _____)

  • A. 1 and 3
  • B. 1 and 5 ✓
  • C. 3 and 5
  • D. 5 and 15

Answer: B — Factors of 15 are 1, 3, 5, 15 and factors of 25 are 1, 5, 25; the common factors are 1 and 5.

Q7. Mowgli jumps by 3 steps starting from 0. Which number will he touch?

  • A. 16
  • B. 18 ✓
  • C. 19
  • D. 17

Answer: B — Jumping by 3 steps from 0 touches 3, 6, 9, 12, 15, 18... so Mowgli lands on 18, which is a multiple of 3.

Q8. Which of the following numbers is a prime number?

  • A. 15
  • B. 21
  • C. 13 ✓
  • D. 12

Answer: C — 13 is a prime number because it has only 2 factors (1 and 13), while 15, 21, and 12 have more than 2 factors.

Q9. A rabbit jumps by 4 and a deer jumps by 6. On which number do they both first land together?

  • A. 10
  • B. 12 ✓
  • C. 18
  • D. 24

Answer: B — The rabbit touches 4, 8, 12, 16... and the deer touches 6, 12, 18... so they first meet at 12, which is the least common multiple.

Q10. In an array of 2 rows with 7 dots in each row, the factors shown are _____ and _____.

  • A. 2 and 14
  • B. 1 and 7
  • C. 2 and 7 ✓
  • D. 7 and 14

Answer: C — An array of 2 rows with 7 dots in each row shows 2 × 7 = 14, so the factors shown are 2 and 7.

Flashcards

What is a factor of a number?

A factor is a number that divides another number completely with no remainder — for example, 3 is a factor of 12 because 3 × 4 = 12.

What is a multiple of a number?

A multiple is a number we get when we multiply a number by another whole number — for example, 15 is a multiple of 3 because 3 × 5 = 15.

What are common factors?

Common factors are numbers that divide two or more numbers exactly — for example, 2 and 4 are common factors of 12 and 20.

What are common multiples?

Common multiples are numbers that are multiples of two or more numbers — for example, 12 is a common multiple of 3 and 4 because both 3 and 4 divide it exactly.

What is a prime number?

A prime number has only two factors: 1 and itself — for example, 13 is prime because only 1 and 13 divide it exactly.

Is 1 a factor of every number?

Yes, 1 is always a factor of every number because any number divided by 1 equals itself.

Is every number a multiple of itself?

Yes, every number is a multiple of itself because the number multiplied by 1 equals itself.

How do you find factors of a number using arrays?

Make rectangular arrangements of dots or objects: if you can arrange a number into equal rows and columns, the row and column sizes are its factors — for example, 12 can be arranged as 3 × 4, so 3 and 4 are factors.

How do you find common factors of two numbers?

List all factors of both numbers, then pick the ones that appear in both lists — for example, factors of 12 are 1, 2, 3, 4, 6, 12 and factors of 18 are 1, 2, 3, 6, 9, 18, so common factors are 1, 2, 3, 6.

How do you find common multiples using a number line?

Mark jumps for both numbers on a number line and see where they land on the same spots — for example, jumping by 3 and jumping by 4 both touch 12, so 12 is a common multiple.

Important Board Questions

What is a factor? Give one example. [1 mark]

A factor divides a number exactly. Use the definition: if a × b = number, then a and b are factors.

Write all the factors of 16. How many factors does 16 have? [2 marks]

Find all numbers that divide 16 exactly: 1 × 16, 2 × 8, 4 × 4. Count them all.

A rabbit jumps by 2 steps and a frog jumps by 3 steps from 0. On a number line, which are the first three common multiples they will touch? Explain how you found them. [3 marks]

Mark jumps for 2: 2, 4, 6, 8, 10, 12... and jumps for 3: 3, 6, 9, 12, 15, 18... Find where they both land (common multiples).

Find all common factors of 24 and 36. Show how each factor divides both numbers exactly. What is the largest common factor? [5 marks]

List all factors of 24 (1, 2, 3, 4, 6, 8, 12, 24) and all factors of 36 (1, 2, 3, 4, 6, 9, 12, 18, 36). Write division checks like 24 ÷ 2 = 12 and 36 ÷ 2 = 18. Pick common ones and find the largest.

True or False: A number can have more factors than multiples. Give a reason for your answer. [2 marks]

Think about 12: its factors are 1, 2, 3, 4, 6, 12 (only 6 factors), but its multiples are 12, 24, 36, 48... (endless multiples).

Assertion-Reason Questions for Factors and Multiples

Q1. Assertion (A):

12 is a multiple of 3.

Reason (R):

Because 3 × 4 = 12, which means 3 is a factor of 12.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. When 3 divides 12 exactly to give 4, then 12 is indeed a multiple of 3, and this is correctly explained by the reason.

Q2. Assertion (A):

13 is a prime number.

Reason (R):

Because 13 is an odd number.

Answer: C — A is true but R is false — The assertion is true (13 has only factors 1 and 13), but the reason is false. A number being odd does not make it prime; for example, 9 is odd but not prime (factors: 1, 3, 9). The correct reason would be that 13 has only 1 and itself as factors.

Q3. Assertion (A):

The common factors of 8 and 12 are 1, 2, and 4.

Reason (R):

Because 1, 2, and 4 are factors of 8, and also 1, 2, and 4 divide 12 completely.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both assertion and reason are true and correctly related. Factors of 8 are 1, 2, 4, 8; factors of 12 are 1, 2, 3, 4, 6, 12; common factors are 1, 2, 4. The reason properly explains why these are common factors.

Unlock more Assertion-Reason questions in Premium — unlock →

Case-Based Questions (CBQ) for Factors and Multiples

Case Study:

Rajesh's mother is preparing for Diwali. She has 12 chocolate pieces and 18 sweet pieces. She wants to make identical gift boxes where each box has the same number of chocolates and the same number of sweets, with no pieces left over. She can use any common factor of 12 and 18 as the number of boxes.

Q1. What are the common factors of 12 and 18?

  • A. 1, 2, 3, 6 ✓
  • B. 1, 2, 4, 6
  • C. 1, 3, 6, 12
  • D. 2, 3, 6, 12

Answer: A

Q2. If Rajesh's mother wants to make the maximum number of boxes, how many boxes should she make?

  • A. 3 boxes
  • B. 6 boxes ✓
  • C. 12 boxes
  • D. 18 boxes

Answer: B

Q3. If she makes 6 boxes, how many chocolates and sweets will be in each box?

  • A. 2 chocolates and 3 sweets ✓
  • B. 12 chocolates and 18 sweets
  • C. 6 chocolates and 6 sweets
  • D. 1 chocolate and 1 sweet

Answer: A

Unlock more case studies in Premium — unlock →

Next chapterMaps and Directions →
Found this useful? Share it with your class group.

You've seen the free preview. Open the full interactive study kit — mind map, instant-graded quiz, progress tracking — free, no sign-in required.

Open Full Study Kit — Free →

Premium unlocks the full Question Bank, Daily Practice Papers, Chapter Quiz and Full Mock Papers for every chapter.