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Prime Time

NCERT Class 6 · Mathematics Based on NCERT Class 6 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter introduces the fundamental concepts of multiples, factors, prime numbers, composite numbers, co-prime numbers, and prime factorisation. It uses engaging games like 'idli-vada' and treasure hunting to help students understand common multiples and common factors. The one big idea a student must walk away with is that every number greater than 1 can be uniquely broken down into a product of prime numbers (prime factorisation), and this helps us check divisibility, find common factors, and determine if numbers are co-prime.
Key Concepts (12)
Multiple
A multiple of a number is the product of that number with any whole number (including 0).
e.g. Multiples of 3 are 3, 6, 9, 12, 15, ...
Common Multiple
A number that is a multiple of two or more numbers is called a common multiple of those numbers.
e.g. 15 is a common multiple of 3 and 5 because 15 is a multiple of both 3 and 5.
Factor
A factor of a number is a whole number that divides the number exactly, leaving no remainder.
e.g. Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
Common Factor
A number that is a factor of two or more numbers is called a common factor of those numbers.
e.g. Common factors of 14 and 36 are 1 and 2.
Prime Number
A number that has exactly two distinct factors: 1 and the number itself.
e.g. 7 is a prime number because its only factors are 1 and 7.
Composite Number
A number that has more than two factors.
e.g. 12 is a composite number because it has factors 1, 2, 3, 4, 6, 12.
Co-prime Numbers
Two numbers are said to be co-prime if they have no common factor other than 1.
e.g. 4 and 9 are co-prime because their only common factor is 1.
Prime Factorisation
Writing a number as a product of its prime factors.
e.g. Prime factorisation of 56 is 2 × 2 × 2 × 7.
Perfect Number
A number for which the sum of all its factors (including the number itself) is equal to twice the number.
e.g. 28 is a perfect number because its factors are 1, 2, 4, 7, 14, 28 and their sum is 56, which is twice 28.
Twin Primes
Pairs of prime numbers that have a difference of 2.
e.g. 3 and 5, 17 and 19 are twin primes.
Sieve of Eratosthenes
A method to find all prime numbers up to a given limit by repeatedly crossing out multiples of primes.
e.g. To find primes up to 100, cross out 1, then circle 2 and cross out all multiples of 2, then circle 3 and cross out all multiples of 3, and so on.
Divisibility Test
A shortcut rule to check if one number is divisible by another without performing long division.
e.g. A number is divisible by 5 if its last digit is 0 or 5.
Formulas & Equations (7)
Prime factorisation of a number is unique (except for the order of factors).
Reads as: Every number greater than 1 has exactly one prime factorisation.
Variables: No variables; it is a fundamental theorem.
Use when: When asked to find the prime factorisation of a number or to check if two numbers are co-prime.
⚠ Students think different factor trees give different prime factorisations, but the set of prime factors (with their counts) is always the same.
If one number is divisible by another, the prime factorisation of the second number is included in the prime factorisation of the first number.
Reads as: For a number A to be divisible by B, every prime factor of B must appear in A at least as many times.
Variables: A = dividend, B = divisor.
Use when: To check divisibility without long division, e.g., 'Is 168 divisible by 12?'
⚠ Students think that if all prime factors of B are present in A, it is always divisible. They forget to check the count (exponent) of each prime.
Divisibility by 2: A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.
Reads as: A number ending in an even digit is divisible by 2.
Variables: Last digit = units digit.
Use when: To quickly check if a number is even or divisible by 2.
⚠ Students think all even numbers are composite, forgetting that 2 is prime.
Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
Reads as: Only the last two digits matter for divisibility by 4.
Variables: Last two digits = tens and units digits.
Use when: To check divisibility by 4 for large numbers.
⚠ Students check only the last digit, which does not work for 4.
Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
Reads as: Numbers ending in 0 or 5 are divisible by 5.
Variables: Last digit = units digit.
Use when: To quickly check divisibility by 5.
⚠ Students think numbers ending in 0 are only divisible by 10, forgetting they are also divisible by 5.
Divisibility by 8: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
Reads as: Only the last three digits matter for divisibility by 8.
Variables: Last three digits = hundreds, tens, and units digits.
Use when: To check divisibility by 8 for large numbers.
⚠ Students try to use the last two digits, which only works for 4, not 8.
Divisibility by 10: A number is divisible by 10 if its last digit is 0.
Reads as: Numbers ending in 0 are divisible by 10.
Variables: Last digit = units digit.
Use when: To quickly check divisibility by 10.
⚠ Students think numbers ending in 5 are also divisible by 10.
How It Works
Finding Common Multiples (Idli-Vada Game)
  1. Step 1: List the multiples of the first number (e.g., multiples of 3: 3, 6, 9, 12, 15, ...).
  2. Step 2: List the multiples of the second number (e.g., multiples of 5: 5, 10, 15, 20, ...).
  3. Step 3: Identify numbers that appear in both lists. These are the common multiples (e.g., 15, 30, 45, ...).
  4. Step 4: The first common multiple is the smallest number that is a multiple of both numbers.
💡 Students often forget that 0 is a multiple of every number, but in the game we start from 1. Also, the first common multiple is not always the product of the two numbers (e.g., for 4 and 6, the first common multiple is 12, not 24).
Finding Common Factors (Treasure Hunting Game)
  1. Step 1: List all factors of the first number (e.g., factors of 14: 1, 2, 7, 14).
  2. Step 2: List all factors of the second number (e.g., factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36).
  3. Step 3: Identify numbers that appear in both lists. These are the common factors (e.g., 1 and 2).
  4. Step 4: The largest common factor is called the Highest Common Factor (HCF) or Greatest Common Divisor (GCD).
💡 Students often confuse common factors with common multiples. Common factors divide the numbers, while common multiples are divided by the numbers.
Prime Factorisation using Factor Tree
  1. Step 1: Write the number at the top of the tree.
  2. Step 2: Break the number into any two factors (e.g., 56 = 4 × 14).
  3. Step 3: Continue breaking each composite factor into smaller factors until all branches end in prime numbers.
  4. Step 4: Collect all the prime numbers at the ends of the branches. Their product is the prime factorisation (e.g., 56 = 2 × 2 × 2 × 7).
💡 Students often stop before all factors are prime. They must keep dividing until every factor is a prime number. Also, different factor trees give the same prime factorisation (order may differ).
Checking if Two Numbers are Co-prime using Prime Factorisation
  1. Step 1: Find the prime factorisation of both numbers (e.g., 56 = 2 × 2 × 2 × 7, 63 = 3 × 3 × 7).
  2. Step 2: Look for any common prime factor (e.g., 7 is common).
  3. Step 3: If there is at least one common prime factor, the numbers are NOT co-prime. If there are no common prime factors, the numbers are co-prime.
💡 Students often think that if two numbers are both composite, they cannot be co-prime. But 8 and 9 are both composite and co-prime because they share no common prime factor.
Sieve of Eratosthenes
  1. Step 1: Write all numbers from 1 to 100 (or any limit).
  2. Step 2: Cross out 1 because it is neither prime nor composite.
  3. Step 3: Circle 2, then cross out all multiples of 2 (4, 6, 8, ...).
  4. Step 4: Circle the next uncrossed number (3), then cross out all its multiples (6, 9, 12, ...).
  5. Step 5: Repeat for 5, 7, 11, etc., until all numbers are either circled (prime) or crossed out (composite).
💡 Students often forget to cross out the multiples of the circled number itself (e.g., after circling 3, they should cross out 6, 9, 12, ... but not 3 itself). Also, they may stop too early.
Exam Traps — Don't Make These Mistakes
⚠ Confusing 'multiple' with 'factor'
✗ Students write 'Factors of 12 are 12, 24, 36, ...'
→ Factors are numbers that divide the given number exactly. Multiples are the products of the number with other whole numbers. 12 is a factor of 12, but 24 and 36 are multiples of 12, not factors.
✓ Factors of 12 are 1, 2, 3, 4, 6, 12. Multiples of 12 are 12, 24, 36, 48, ...
⚠ Thinking 1 is a prime number
✗ Students write '1 is a prime number because it has only one factor.'
→ A prime number must have exactly two distinct factors: 1 and itself. 1 has only one factor (1), so it is neither prime nor composite.
✓ 1 is neither a prime number nor a composite number.
⚠ Assuming all even numbers are composite
✗ Students write 'All even numbers are composite numbers.'
→ The number 2 is even and has only two factors (1 and 2), so it is prime. All other even numbers are composite because they have at least 2 as a factor.
✓ All even numbers greater than 2 are composite. 2 is the only even prime number.
⚠ Incorrectly checking co-prime by looking at any factorisation
✗ Students write '56 and 63 are co-prime because 56 = 14 × 4 and 63 = 21 × 3, and there are no common factors.'
→ This factorisation does not show all factors. Using prime factorisation (56 = 2×2×2×7, 63 = 3×3×7) reveals that 7 is a common factor, so they are not co-prime.
✓ To check if two numbers are co-prime, find their prime factorisations. If there is no common prime factor, they are co-prime.
⚠ Divisibility by 4: checking only the last digit
✗ Students write '22 is divisible by 4 because its last digit is 2.'
→ Divisibility by 4 depends on the last two digits, not just the last digit. 22 is not divisible by 4 because 22 ÷ 4 = 5 remainder 2.
✓ A number is divisible by 4 if the number formed by its last two digits is divisible by 4. For example, 124 is divisible by 4 because 24 is divisible by 4.
Compare & Contrast
Number of factors
Prime Number
Composite Number
A prime number has exactly two factors (1 and itself), while a composite number has more than two factors.
Relationship to the given number
Factor
Multiple
A factor divides the number exactly (e.g., 3 is a factor of 12). A multiple is the product of the number with another whole number (e.g., 12 is a multiple of 3).
Definition
Common Factor
Common Multiple
A common factor is a number that divides two or more numbers exactly. A common multiple is a number that is a multiple of two or more numbers.
Exam Focus
Marks pattern: This chapter typically appears with 2-3 MCQs (1 mark each), 1-2 short answer questions (2-3 marks), and 1 long answer question (5 marks) on prime factorisation or divisibility.
Most asked: Prime factorisation of numbers · Identifying prime and composite numbers · Finding common factors and common multiples · Checking if numbers are co-prime · Divisibility tests for 2, 4, 5, 8, 10
Don't skip: Prime factorisation of a given number (e.g., 1728, 1000) — often a 5-mark question.; Word problems involving common multiples or common factors (e.g., idli-vada game, treasure hunting).; Checking if two numbers are co-prime using prime factorisation.
One-Line Revision (read aloud before the exam)
  1. A multiple of a number is the product of that number with any whole number.
  2. A factor of a number divides the number exactly, leaving no remainder.
  3. Common multiples are numbers that are multiples of two or more numbers.
  4. Common factors are numbers that divide two or more numbers exactly.
  5. A prime number has exactly two factors: 1 and itself.
  6. A composite number has more than two factors.
  7. 1 is neither prime nor composite.
  8. Two numbers are co-prime if they have no common factor other than 1.
  9. Prime factorisation is writing a number as a product of its prime factors.
  10. Every number greater than 1 has a unique prime factorisation.
  11. A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.
  12. A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
  13. A number is divisible by 5 if its last digit is 0 or 5.
  14. A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
  15. A number is divisible by 10 if its last digit is 0.
  16. The Sieve of Eratosthenes is a method to find prime numbers up to a given limit.
  17. Twin primes are pairs of primes with a difference of 2, like 3 and 5.
  18. A perfect number is one where the sum of its factors equals twice the number, like 6 and 28.

MCQs — 10 Questions with Answers

Q1. Which of these is a multiple of 5?

  • A. 23
  • B. 35 ✓
  • C. 42
  • D. 51

Answer: B — 35 = 5 × 7, so it is a multiple of 5; the others when divided by 5 leave a remainder.

Q2. What are all the factors of 12?

  • A. 1, 2, 3, 4, 6, 12 ✓
  • B. 1, 2, 4, 8, 12
  • C. 2, 3, 4, 6
  • D. 1, 3, 4, 6, 12

Answer: A — All numbers that divide 12 exactly with no remainder are 1, 2, 3, 4, 6, and 12.

Q3. Which number is prime?

  • A. 1
  • B. 15
  • C. 17 ✓
  • D. 20

Answer: C — 17 has exactly 2 factors (1 and 17); 1 has only one factor, 15 and 20 have many factors.

Q4. In the idli-vada game up to 30, how many times do you say 'idli-vada'?

  • A. 1 time
  • B. 2 times ✓
  • C. 3 times
  • D. 5 times

Answer: B — Common multiples of 3 and 5 up to 30 are 15 and 30, so 'idli-vada' is said 2 times.

Q5. What is the smallest common multiple of 4 and 6?

  • A. 4
  • B. 6
  • C. 12 ✓
  • D. 24

Answer: C — 12 is the first number that appears in both multiples of 4 (4, 8, 12...) and multiples of 6 (6, 12...).

Q6. Which of the following is a composite number?

  • A. 2
  • B. 13
  • C. 18 ✓
  • D. 19

Answer: C — 18 has factors 1, 2, 3, 6, 9, 18 (more than 2 factors); 2, 13, 19 are all prime numbers.

Q7. In Jump Jackpot, which jump size will reach both 20 and 30?

  • A. 6
  • B. 10 ✓
  • C. 15
  • D. 25

Answer: B — 10 is a common factor of 20 (20 = 10 × 2) and 30 (30 = 10 × 3); other options don't divide both numbers.

Q8. Which statement is TRUE?

  • A. 1 is a prime number
  • B. All prime numbers are odd
  • C. 2 is the only even prime number ✓
  • D. Every composite number has an even number of factors

Answer: C — 2 is prime with factors 1 and 2; it is the only even prime because all other even numbers have 2 as a factor.

Q9. __________ are numbers that have exactly two factors: 1 and the number itself.

  • A. Composite numbers
  • B. Prime numbers ✓
  • C. Multiples
  • D. Common factors

Answer: B — By definition, prime numbers have exactly 2 factors; composite numbers have more than 2 factors.

Q10. Look at the Venn diagram of multiples of 3 and 5 up to 30. Which number appears in the overlapping region?

  • A. 9
  • B. 10
  • C. 15 ✓
  • D. 20

Answer: C — 15 is a multiple of both 3 and 5 (15 = 3 × 5), so it is in the common region; others are multiples of only one.

Flashcards

What are multiples of 3?

Numbers you get by skip-counting by 3: 3, 6, 9, 12, 15, 18, and so on.

What is a factor of 24?

A factor is any number that divides 24 exactly with no remainder: 1, 2, 3, 4, 6, 8, 12, and 24.

What are common multiples?

Numbers that appear in the multiple list of two or more numbers, like 15, 30, 45 for both 3 and 5.

What are common factors?

Numbers that divide two or more numbers exactly, like 1 and 2 are common factors of 14 and 36.

Define a prime number.

A number with exactly 2 factors: 1 and itself, like 2, 3, 5, 7, 11, 13, 17, 19.

Define a composite number.

A number with more than 2 factors, like 4, 6, 8, 9, 10, 12 that can be arranged in different rectangular ways.

Why is 1 not prime or composite?

The number 1 has only one factor (itself), but prime and composite numbers must have at least 2 or more than 2 factors.

In the idli-vada game, when do you say 'idli-vada'?

Say 'idli-vada' for numbers that are multiples of BOTH 3 and 5, like 15, 30, 45, 60.

What jump sizes will land on 36 in Jump Jackpot?

Any factor of 36 works as a jump size: 1, 2, 3, 4, 6, 9, 12, 18, or 36.

Which is the smallest common multiple of 3 and 5?

The smallest common multiple of 3 and 5 is 15 (which equals 3 × 5).

Important Board Questions

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

A multiple is a number made by skip-counting by that number. For example, multiples of 4 are 4, 8, 12, 16...

Find all common factors of 12 and 18. [2 marks]

List all factors of 12 (1, 2, 3, 4, 6, 12) and all factors of 18 (1, 2, 3, 6, 9, 18). Then circle the numbers that appear in BOTH lists.

In the idli-vada game, explain with an example when a player should say 'idli-vada' instead of a number. Why do we call these numbers common multiples? [3 marks]

A player says 'idli-vada' when the number is a multiple of BOTH 3 and 5 (like 15). These are called common multiples because 15 is in BOTH the list of multiples of 3 AND the list of multiples of 5.

Guna wants to arrange 12 mangoes and Anshu wants to arrange 7 mangoes in rectangular rows and columns. (a) In how many ways can Guna arrange 12 mangoes? (b) In how many ways can Anshu arrange 7 mangoes? (c) Explain why Guna has more arrangements than Anshu using the words 'factors' or 'prime number' or 'composite number.' [5 marks]

For Guna: find all factors of 12 (1×12, 2×6, 3×4) = 3 ways. For Anshu: find all factors of 7 (1×7 only) = 1 way. Explain: 12 is composite (has many factors), so many rectangle arrangements. 7 is prime (has only 2 factors: 1 and 7), so only 1 rectangle arrangement.

True or False: (a) The number 2 is a prime number. (b) Every composite number can be arranged in more than one rectangular way. Give a reason for each. [2 marks]

(a) True: 2 has exactly 2 factors (1 and 2). (b) True: Composite numbers have more than 2 factors, so they can form different rectangle arrangements like 12 = 2×6, 3×4, 1×12.

Assertion-Reason Questions for Prime Time

Q1. Assertion (A):

The number 15 is both a multiple of 3 and a multiple of 5.

Reason (R):

15 = 3 × 5, so 15 is divisible by both 3 and 5 with no remainder.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains why 15 is a multiple of both 3 and 5. This makes it a common multiple.

Q2. Assertion (A):

All prime numbers are odd numbers.

Reason (R):

Prime numbers have exactly two factors: 1 and themselves.

Answer: D — A is false but R is true — The assertion is false because 2 is a prime number that is even. The reason is true (it correctly defines prime numbers), but it doesn't explain why the assertion is false.

Q3. Assertion (A):

A factor of a number always divides that number exactly with no remainder.

Reason (R):

Factors are numbers that can be arranged in rectangular arrays.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — Both statements are true. The assertion correctly defines what a factor is. The reason is also true (as shown in the fig arrangement activity), but it explains the same idea differently rather than being the cause of the assertion.

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Case-Based Questions (CBQ) for Prime Time

Case Study:

Priya's mother runs a small sweet shop. She makes two types of sweets: ladoos and barfis. Every day, she makes 12 ladoos and 18 barfis. She wants to pack them in identical gift boxes such that each box has the same number of ladoos and the same number of barfis, with no sweets left over.

Q1. What is the maximum number of identical gift boxes Priya's mother can make?

  • A. 6 boxes ✓
  • B. 8 boxes
  • C. 10 boxes
  • D. 12 boxes

Answer: A

Q2. How many ladoos and how many barfis will be in each gift box?

  • A. 2 ladoos and 3 barfis ✓
  • B. 3 ladoos and 2 barfis
  • C. 4 ladoos and 6 barfis
  • D. 6 ladoos and 9 barfis

Answer: A

Q3. If Priya's mother decides to make 24 ladoos and 36 barfis instead, how would the maximum number of boxes change?

  • A. It would decrease to 4 boxes
  • B. It would increase to 12 boxes ✓
  • C. It would remain 6 boxes
  • D. It would increase to 8 boxes

Answer: B

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