This MCQ module is based on: 5.2 Prime Numbers
TOPIC 17 OF 41
5.2 Prime Numbers
🎓 Class 6
Mathematics
CBSE
Theory
Ch 5 — Prime Time
⏱ ~15 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: 5.2 Prime Numbers
Targeting Class 6 level in Number Theory, with Basic difficulty.
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5.2 Prime Numbers
Some numbers have very few factors — only two of them: 1 and the number itself. These are called prime numbers?. Numbers that have more than two factors are called composite numbers?.
Definition
A whole number greater than 1 is prime if its only factors are 1 and itself. Otherwise (if it has three or more factors) it is composite. The number 1 is neither prime nor composite — it has only one factor.
Guna's Number Game
Guna writes six numbers on a board: 4, 16, 25, 43. For each, the students share a feature:
- Karnavati: "9 is special because it is a single-digit number whereas all the other numbers are 2-digit numbers."
- Gurupreet: "9 is special because it is a multiple of 3 while the others are not."
- Murugan: "16 is special because it is the only multiple of 2 (or 4)."
- Gopika: "25 is special because it is the only multiple of 5."
- Tadnyikee: "43 is special because it is the only prime number."
- Radha: "43 is special because it is the only number that is not a square."
Recognising Primes — a Closer Look
Consider single-digit numbers:
- Factors of 2: 1, 2 → prime
- Factors of 3: 1, 3 → prime
- Factors of 4: 1, 2, 4 → composite
- Factors of 5: 1, 5 → prime
- Factors of 6: 1, 2, 3, 6 → composite
- Factors of 7: 1, 7 → prime
- Factors of 8: 1, 2, 4, 8 → composite
- Factors of 9: 1, 3, 9 → composite
🔵 Which of these could be the other number: 2, 3, 5, 8, 10? Among 2, 3, 5 — primes; 8, 10 — composite. The "special" number in such context is usually the prime.
The Sieve of Eratosthenes — Finding All Primes up to 100
An ancient Greek mathematician, Eratosthenes of Cyrene (276–194 BCE), invented a beautiful method (a "sieve") to catch all primes in a range. Here is the algorithm:
- Write numbers 1 to 100 in a 10-column grid.
- Cross out 1 (not prime).
- Circle 2; then cross out every other multiple of 2 (4, 6, 8, …).
- Circle 3; cross out every other multiple of 3 (6, 9, 12, …).
- Circle 5; cross out every other multiple of 5.
- Circle 7; cross out every other multiple of 7.
- All remaining uncrossed numbers are primes.
Sieve of Eratosthenes — red cells mark the 25 primes less than 100.
The 25 primes less than 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
Historical Note
Euclid (c. 300 BCE) proved in Book IX of his Elements that there are infinitely many primes. The proof is one of the most elegant in mathematics: if there were only finitely many primes, multiplying them all together and adding 1 produces a number that cannot be divided by any of those primes — a contradiction.
5.3 Co-prime Numbers
Two numbers are co-prime? if their only common factor is 1. They need not be prime themselves — for example, 4 and 9 are both composite, yet they share no factor except 1, so they are co-prime.
Example: Are 15 and 39 co-prime? Factors of 15: 1, 3, 5, 15. Factors of 39: 1, 3, 13, 39. Common factor other than 1 exists (it is 3), so 15 and 39 are not co-prime. But 4 and 15 share only 1 — they ARE co-prime.
Which of the following pairs of numbers are co-prime?
(a) 18 and 35 (b) 15 and 37 (c) 30 and 415 (d) 17 and 69 (e) 81 and 18
(a) 18 and 35 (b) 15 and 37 (c) 30 and 415 (d) 17 and 69 (e) 81 and 18
(a) 18 = 2·3², 35 = 5·7. No common prime → co-prime. (b) 15 = 3·5, 37 is prime. No common → co-prime. (c) 30 = 2·3·5, 415 = 5·83. Common 5 → not co-prime. (d) 17 is prime, 69 = 3·23. No common → co-prime. (e) 81 = 3⁴, 18 = 2·3². Common 3 → not co-prime.
Co-prime Art — The Thread Diagram
Place 10 pegs in a circle, numbered 0 to 9. Wind a thread starting at 0, jumping forward by 3 each time (so 0→3→6→9→2→5→8→1→4→7→0). The thread visits every peg exactly once because 3 and 10 are co-prime! With 10 pegs and a jump of 4 (which is not co-prime with 10), the thread visits only half the pegs: 0→4→8→2→6→0.
Thread diagrams. When jump size and peg count are co-prime, every peg is visited.
Activity: Sieve of Eratosthenes (Hands-on)
L3 Apply
Materials: Squared paper, 4 coloured pencils (yellow, red, green, blue), pencil
Predict: How many primes will remain below 50? Write your guess.
- Draw a 10×10 grid. Write 1 to 100, one per cell.
- Cross out 1 with pencil (it's neither prime nor composite).
- Circle 2 in yellow. Cross out every 2nd number after it (4, 6, …).
- Circle 3 in red. Cross out every 3rd number (6 is already crossed, then 9, 12, …).
- Circle 5 in green. Cross out every 5th.
- Circle 7 in blue. Cross out every 7th (14, 21, …).
- Count circled/uncrossed numbers. You should find 25 primes.
Primes below 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 → 15 primes.
Competency-Based Questions
Scenario: Priya is labelling file folders with numbers. She wants to mark folders labelled with a prime number with a star, and folders with co-prime pair labels (e.g. with their neighbour) with a ribbon.
Q1. Among her folders 51, 53, 57, 59, 61, which get stars?
L3 Apply51 = 3·17 (not), 53 prime, 57 = 3·19 (not), 59 prime, 61 prime → stars on 53, 59, 61.
Q2. Classify: which of these pairs share a ribbon (are co-prime)? (12, 35), (14, 21), (25, 28).
L4 Analyse(12, 35): 12 = 2²·3, 35 = 5·7 → co-prime. (14, 21): share 7 → not. (25, 28): 25 = 5², 28 = 2²·7 → co-prime.
Q3. Priya claims: "If two numbers are both prime and different, they must be co-prime." Evaluate.
L5 EvaluateTrue. A prime has only 1 and itself as factors; two DIFFERENT primes have no factor in common except 1.
Q4. Design a thread-diagram challenge: choose a peg count \(n\) and a jump size \(j\) so that exactly half the pegs are visited. Justify.
L6 CreateChoose \(n=12\), \(j=2\). HCF(12, 2) = 2, so number of pegs visited = n/HCF = 6, exactly half. Any pair where HCF = 2 and n is even works.
Assertion–Reason Questions
A: 2 is the only even prime number.
R: Every even number greater than 2 has 2 as a factor in addition to 1 and itself.
R: Every even number greater than 2 has 2 as a factor in addition to 1 and itself.
Answer: (a) — R directly explains why A holds.
A: Every pair of co-prime numbers consists of two primes.
R: 4 and 9 are co-prime.
R: 4 and 9 are co-prime.
Answer: (d) — A is false (4 and 9 are composite yet co-prime). R is a true counter-example.
A: 1 is a prime number.
R: 1 divides every whole number.
R: 1 divides every whole number.
Answer: (d) — A false: 1 has only one factor, so it is neither prime nor composite. R is true but unrelated.
Frequently Asked Questions — Prime Time
What is Part 2 — Prime Numbers, Co-primes & the Sieve of Eratosthenes | Class 6 Maths | MyAiSchool in NCERT Class 6 Mathematics?
Part 2 — Prime Numbers, Co-primes & the Sieve of Eratosthenes | Class 6 Maths | MyAiSchool is a key concept covered in NCERT Class 6 Mathematics, Chapter 5: Prime Time. This lesson builds the student's foundation in the chapter by explaining the core ideas with worked examples, definitions, and step-by-step methods aligned to the CBSE curriculum.
How do I solve problems on Part 2 — Prime Numbers, Co-primes & the Sieve of Eratosthenes | Class 6 Maths | MyAiSchool step by step?
To solve problems on Part 2 — Prime Numbers, Co-primes & the Sieve of Eratosthenes | Class 6 Maths | MyAiSchool, follow the NCERT method: identify the given quantities, choose the relevant formula or theorem, substitute values carefully, and simplify. Class 6 exercises gradually increase in difficulty — start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.
What are the most important formulas for Chapter 5: Prime Time?
The essential formulas of Chapter 5 (Prime Time) are listed in the chapter summary and highlighted throughout the lesson in formula boxes. Memorise them and practise at least 2–3 problems per formula. CBSE board exams frequently test direct application as well as combined use of multiple formulas from this chapter.
Is Part 2 — Prime Numbers, Co-primes & the Sieve of Eratosthenes | Class 6 Maths | MyAiSchool important for the Class 6 board exam?
Part 2 — Prime Numbers, Co-primes & the Sieve of Eratosthenes | Class 6 Maths | MyAiSchool is part of the NCERT Class 6 Mathematics syllabus and appears in CBSE board exams. Questions typically include short-answer, long-answer, and competency-based items. Review the NCERT examples, exercise questions, and previous-year board problems on this topic to prepare confidently.
What mistakes should students avoid in Part 2 — Prime Numbers, Co-primes & the Sieve of Eratosthenes | Class 6 Maths | MyAiSchool?
Common mistakes in Part 2 — Prime Numbers, Co-primes & the Sieve of Eratosthenes | Class 6 Maths | MyAiSchool include skipping steps, misapplying formulas, sign errors, and losing track of units. Write each step clearly, double-check algebraic manipulations, and re-read the question after solving to verify that your answer matches what was asked.
Where can I find more NCERT practice questions on Part 2 — Prime Numbers, Co-primes & the Sieve of Eratosthenes | Class 6 Maths | MyAiSchool?
End-of-chapter NCERT exercises for Part 2 — Prime Numbers, Co-primes & the Sieve of Eratosthenes | Class 6 Maths | MyAiSchool cover all difficulty levels tested in CBSE exams. After completing them, try the examples again without looking at the solutions, attempt the NCERT Exemplar questions for Chapter 5, and solve at least one previous-year board paper to consolidate your understanding.
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