Prime Time – Class 6 Maths Chapter 5 Notes & NCERT PDF

Class 6MathematicsChapter 5NCERT book: Ganita Prakash - Mathematics Class 6

Notes and a simple summary of Chapter 5 of the NCERT Class 6 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 5: Prime Time

Introduction

Chapter 5 of Ganita Prakash is about the "building blocks" of numbers. It starts with a playground game, Idli-Vada, and a treasure game, Jump Jackpot, and uses them to lead us to multiples, factors, primes, co-primes, prime factorisation and divisibility tests. The friends Guna and Anshu help us along the way.

Prime numbers have exactly two factors: 1 and the number itself. Composite numbers have more than two factors. 1 is neither prime nor composite.

Key Ideas

5.1 Common multiples and common factors

The Idli-Vada game: children count 1, 2, 3, … in a circle. On multiples of 3 they say idli, on multiples of 5 they say vada, and on numbers that are multiples of both they say idli-vada. The first idli-vada comes at 15. Numbers like 15, 30, 45, … are common multiples of 3 and 5.

From 1 to 90: idli is said 30 times, vada 18 times and idli-vada 6 times.

The 10th idli-vada is said at 150.

Jump Jackpot: Grumpy puts a treasure on a number, and Jumpy picks a jump size starting from 0. For a treasure on 24, the jump sizes that work are exactly the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.

With two treasures, only the common factors of both numbers work. For treasures on 14 and 36, only jumps of 1 or 2 land on both.

A perfect number is one whose factors add up to twice the number. 28 is perfect (1 + 2 + 4 + 7 + 14 + 28 = 56), and so is 6 (1 + 2 + 3 + 6 = 12).

5.2 Prime numbers

Guna wants to pack 12 figs (anjeer) in a box, and Anshu wants to pack 7. Twelve figs can be arranged in rectangles in several ways (1 × 12, 2 × 6, 3 × 4). Seven figs fit only as 1 × 7. That is because 7 has just two factors, so 7 is prime.

Sieve of Eratosthenes: to list the primes up to 100:

Cross out 1.

Circle 2 and cross out every later multiple of 2.

Circle the next uncrossed number (3) and cross out its later multiples.

Repeat with 5, 7, … until every number is circled or crossed out.

The circled numbers are the primes. There are 25 primes up to 100. Eratosthenes was a Greek mathematician who lived about 2200 years ago.

2 is the only even prime.

Twin primes differ by 2. Below 100 they are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61) and (71, 73).

90 to 96 are seven consecutive composite numbers. The biggest gap between two primes below 100 is 97 − 89 = 8.

5.3 Co-prime numbers for safekeeping treasures

In this version of the game a jump of 1 is not allowed. A pair like 4 and 9 is "safe" because no jump size other than 1 reaches both. Co-prime numbers have no common factor other than 1.

15 and 39 are not co-prime (3 divides both).

18 and 35, 15 and 37, and 17 and 69 are co-prime.

In the thread-art activity, the thread reaches every peg only when the number of pegs and the thread-gap are co-prime.

5.4 Prime factorisation

Keep breaking a composite number into factors until only primes are left. For example, 56 = 4 × 14 = 2 × 2 × 2 × 7.

Every number greater than 1 has exactly one prime factorisation, apart from the order of the factors.

For 36, all four ways of breaking it down (2 × 18, 3 × 12, 4 × 9, 6 × 6) end at two 2s and two 3s.

Checking co-primes: if two numbers share no prime factor, they are co-prime. Anshu thought 56 = 14 × 4 and 63 = 21 × 3 share no factor, but Guna spotted 7 in both: 56 = 2 × 2 × 2 × 7 and 63 = 3 × 3 × 7. On the other hand, 40 = 2 × 2 × 2 × 5 and 231 = 3 × 7 × 11 are co-prime.

Checking divisibility: a number is divisible by another if the second number's prime factorisation is contained in the first's.

- 168 = 2 × 2 × 2 × 3 × 7 contains 12 = 2 × 2 × 3, so 168 = 12 × 14.

- 42 = 2 × 3 × 7 has only one 2, so it is not divisible by 12, which needs two 2s.

5.5 Divisibility tests

10: the number ends in 0.

5: the number ends in 0 or 5.

2: the number ends in 0, 2, 4, 6 or 8.

4: check the number formed by the last two digits. 8536 works because 36 is divisible by 4. The last digit alone is not enough: 12 is a multiple of 4 but 22 is not.

8: check the number formed by the last three digits.

5.6 Fun with numbers

Special numbers: in the box 9, 16, 25, 43, each number is special for a different reason. For example, 43 is the only prime and 16 is the only even number.

Prime puzzle: fill a grid with primes so that each row and column multiplies to the number shown.

Worked Examples

How many leap years from 2024 to 2099? Take the multiples of 4: 2024, 2028, …, 2096. That is (2096 − 2024) ÷ 4 + 1 = 19. None of them is a century year, so none is skipped.

Treasures on 28 and 70. Factors of 28: 1, 2, 4, 7, 14, 28. Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70. The common factors, and so the jump sizes that work, are 1, 2, 7 and 14.

Three primes below 30 whose product is 1955. 1955 ends in 5, so divide by 5 to get 391. Then 391 = 17 × 23. So 1955 = 5 × 17 × 23.

Smallest multiple of every number from 1 to 10. Use the largest powers of each prime needed: 2 × 2 × 2 (for 8), 3 × 3 (for 9), 5 and 7. So 8 × 9 × 5 × 7 = 2520. Leaving out 7 gives 360.

Is 75 divisible by 21? 75 = 3 × 5 × 5 and 21 = 3 × 7. Since 7 is not a prime factor of 75, the answer is no.

Figure it Out: Hints

For "Who am I?" questions, list the multiples of the given factor first, then test the digit clue. For example, the multiples of 7 below 40 whose digits add to 8 give 35.

To check whether a number is prime, you only need to try dividing by primes up to about its square root.

Guna says "any two prime numbers are co-prime". Test this with pairs like (2, 3) and (3, 11). It is true, because two different primes have no common factor except 1.

Maths Words

Multiple (गुणज)

Factor / divisor (गुणनखंड / भाजक)

Common multiple (सार्व गुणज)

Common factor (सार्व गुणनखंड)

Prime number (अभाज्य संख्या)

Composite number (भाज्य संख्या)

Co-prime (सह-अभाज्य)

Twin primes (युग्म अभाज्य)

Prime factorisation (अभाज्य गुणनखंडन)

Divisibility test (विभाज्यता की जाँच)

Perfect number (पूर्ण संख्या)

Sieve (छलनी)

Historical Connections

Eratosthenes (Greece, about 2200 years ago) invented the sieve for finding primes.

Euclid discovered whether the list of primes ever ends, which you will study in a later class.

The largest prime anyone has "written down" would fill about 6500 pages, so it exists only on a computer.

Common Mistakes

Calling 1 a prime number. It has only one factor, so it is neither prime nor composite.

Thinking all odd numbers are prime. 9, 15, 21 and 51 are all composite (51 = 3 × 17).

Deciding two numbers are co-prime from just one way of factorising them, like Anshu's mistake with 56 and 63. Always use the prime factorisation.

Testing divisibility by 4 using only the last digit. Use the last two digits.

Forgetting repeated primes. 42 has all of 12's prime factors, but only one 2, so 12 does not divide it.

Maths in Daily Life

Common multiples tell you when two repeating events happen together, like two buses that leave every 3 and 5 minutes.

Factors help you arrange things in equal rows: packing boxes, planting trees, seating guests.

Leap years follow a divisibility rule.

Prime numbers keep online banking and messages secure (you will learn more about this later).

💡 Key Learning Points

✓Find common multiples and common factors using the Idli-Vada and Jump Jackpot games
✓Tell prime numbers (exactly two factors) from composite numbers, and know that 1 is neither
✓List primes up to 100 with the Sieve of Eratosthenes and spot twin primes
✓Write the prime factorisation of a number and use it to test co-primes and divisibility
✓Use shortcut divisibility tests for 2, 4, 5, 8 and 10 without long division

👨‍🏫 Teaching Tips

→Play the Idli-Vada game in class with 3 and 5, then with 2 and 5, 3 and 7, and 4 and 6. Ask which pairs make idli-vada appear earlier than their product.
→Draw a number line on the floor for Jump Jackpot and let children jump with chosen step sizes to find common factors.
→Give each group 12 counters and 7 counters to arrange in rectangles, so they see why 7 is prime.
→Run the Sieve of Eratosthenes on a printed 1–100 chart with two colours, then discuss why the method works.
→Build factor trees for 36 in four different ways on the board to show that the prime factorisation is always the same.
→Recreate the thread-art activity with pegs on a circle (or dots on paper) to link full coverage with co-prime numbers.
→Hold a divisibility-rule relay where teams check numbers like 8560 and 14560 for 2, 4, 5, 8 and 10 as fast as possible.

📋 Assessment Questions

Can the student list all common factors of two numbers such as 28 and 70?
Can the student explain why 1 is neither prime nor composite, and why 2 is the only even prime?
Can the student write a prime factorisation (e.g., 1955 = 5 × 17 × 23) and use it to decide whether two numbers are co-prime?
Can the student use the last-two-digits rule to test divisibility by 4 and the last-three-digits rule for 8?
Can the student find the smallest number divisible by all of 1 to 10?

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