Class 9 Mathematics Chapter 5 Prime Time Previous Year Questions (2020–2025) with Solutions

Reading your textbook teaches you concepts; solving past papers teaches you how examiners test them. A 5-mark question on HCF and LCM isn't just computation—it's a two-step word problem where you must first model the scenario, then choose the right method. CBSE changes its emphasis every 2–3 years: some cycles stress divisibility tests; others lean heavy on prime factorisation and real-world applications. By working through 2020–2025 papers, you'll spot that 3-mark questions almost always pair HCF with real-world contexts (e.g., arranging sweets into boxes), while 1-mark questions test pure definitional recall or quick divisibility checks. Past papers also build confidence: you'll realise 60–70% of questions follow predictable templates. This means less panic on exam day and more room for strategic thinking. Additionally, timing yourself on these papers reveals whether you can solve HCF/LCM in under 4 minutes—a critical skill when you have 25 questions in 90 minutes.

One-mark questions in Prime Time test rapid recall and basic divisibility logic. These five are the most common variations: **Q1:** Which of the following is a prime number? (A) 1 (B) 21 (C) 29 (D) 51 **Answer: (C) 29.** Prime numbers have exactly two factors: 1 and themselves. 29's only divisors are 1 and 29. [1 is not prime; 21 = 3 × 7; 51 = 3 × 17.] **Q2:** A number is divisible by 9 if: (A) It is divisible by 3 (B) The sum of its digits is divisible by 9 (C) Its last digit is 9 (D) It is an odd number **Answer: (B).** The divisibility rule for 9 states the sum of all digits must be divisible by 9. Example: 738 → 7 + 3 + 8 = 18, and 18 ÷ 9 = 2, so 738 is divisible by 9. **Q3:** What is the HCF of 24 and 36? (A) 6 (B) 8 (C) 12 (D) 24 **Answer: (C) 12.** Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Highest common is 12. **Q4:** The LCM of 5 and 7 is: (A) 12 (B) 35 (C) 25 (D) 1 **Answer: (B) 35.** Since 5 and 7 are both prime, LCM = 5 × 7 = 35. **Q5:** Which number is composite? (A) 17 (B) 19 (C) 23 (D) 25 **Answer: (D) 25.** Composite numbers have more than two factors. 25 = 5 × 5, so factors are 1, 5, 25. [17, 19, 23 are all prime.]

Three-mark questions require two steps: identifying the method and executing calculations. These are real-paper patterns: **Q1: Find the HCF and LCM of 60 and 90.** *Solution:* Prime factorisation: 60 = 2² × 3 × 5 90 = 2 × 3² × 5 HCF = product of smallest powers of common primes = 2¹ × 3¹ × 5¹ = 30. LCM = product of highest powers of all primes = 2² × 3² × 5 = 4 × 9 × 5 = 180. **Verify:** HCF × LCM = 30 × 180 = 5400, and 60 × 90 = 5400. ✓ **Q2: A sweet shop wants to arrange 144 gulab jamuns and 96 laddus in identical gift boxes such that each box has the same number of gulab jamuns and laddus. What is the maximum number of boxes?** *Solution:* Maximum boxes = HCF(144, 96). 144 = 2⁴ × 3² 96 = 2⁵ × 3 HCF = 2⁴ × 3 = 16 × 3 = 48 boxes. Each box will have 144 ÷ 48 = 3 gulab jamuns and 96 ÷ 48 = 2 laddus. **Q3: Use the Sieve of Eratosthenes to find all prime numbers between 10 and 30.** *Solution:* Write 10–30: 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30. Strike multiples of 2 (except 2): 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30 ✗ Strike multiples of 3 (except 3): 15, 21, 27 ✗ Strike multiples of 5 (except 5): 25 ✗ Remaining: **11, 13, 17, 19, 23, 29** are primes. **Q4: Check if 2156 is divisible by 11 using the divisibility test.** *Solution:* Divisibility by 11: (Sum of digits at odd places) − (Sum of digits at even places) must be divisible by 11. 2156: Odd places (right to left): 6, 1 → sum = 7. Even places: 5, 2 → sum = 7. (7 − 7) = 0, which is divisible by 11. So 2156 is divisible by 11. Verify: 2156 ÷ 11 = 196. ✓ **Q5: Find the smallest number divisible by 12, 15, and 18.** *Solution:* Required number = LCM(12, 15, 18). 12 = 2² × 3 15 = 3 × 5 18 = 2 × 3² LCM = 2² × 3² × 5 = 4 × 9 × 5 = **180**.

Five-mark questions test deeper understanding: multi-step reasoning, verification, and application. These three are the most common board patterns: **Q1: Two bells ring at intervals of 20 minutes and 30 minutes respectively. They rang together at 8:00 AM. At what time will they ring together again? Also, how many times will they ring together in a 5-hour period starting from 8:00 AM?** *Solution:* The bells will ring together at intervals = LCM(20, 30). 20 = 2² × 5 30 = 2 × 3 × 5 LCM = 2² × 3 × 5 = 60 minutes. They rang together at 8:00 AM. Next time = 8:00 AM + 60 minutes = **9:00 AM**. In 5 hours (300 minutes): Number of intervals = 300 ÷ 60 = 5. They ring together at: 8:00, 9:00, 10:00, 11:00, 12:00, 1:00 PM → **6 times** (including initial). **Q2: Find the HCF of 144, 192, and 240 using the prime factorisation method. Verify your answer using the Euclidean algorithm for at least one pair.** *Solution:* Prime factorisation: 144 = 2⁴ × 3² 192 = 2⁶ × 3 240 = 2⁴ × 3 × 5 HCF = 2⁴ × 3 = 16 × 3 = **48**. Verification using Euclidean algorithm for 144 and 192: 192 = 144 × 1 + 48 144 = 48 × 3 + 0 So HCF(144, 192) = 48. ✓ Now HCF(48, 240): 240 = 48 × 5 + 0 So HCF(48, 240) = 48. ✓ Therefore, HCF(144, 192, 240) = **48**. **Q3: A shopkeeper has 105 lemons, 84 oranges, and 147 guavas. He wants to distribute them equally into gift hampers such that no fruit is left. (i) Find the maximum number of hampers. (ii) How many of each fruit will be in each hamper? (iii) Verify that the total number of fruits is correctly distributed.** *Solution:* (i) Maximum hampers = HCF(105, 84, 147). 105 = 3 × 5 × 7 84 = 2² × 3 × 7 147 = 3 × 7² HCF = 3 × 7 = **21 hampers**. (ii) Fruits per hamper: • Lemons: 105 ÷ 21 = 5 • Oranges: 84 ÷ 21 = 4 • Guavas: 147 ÷ 21 = 7 (iii) Verification: Total fruits = 105 + 84 + 147 = 336. Distributed total = 21 × (5 + 4 + 7) = 21 × 16 = 336. ✓

The CBSE curriculum revision (2024–25 onwards) has subtly shifted emphasis in Prime Time. Historically, boards over-tested pure HCF/LCM computation, but the new pattern (visible in recent sample papers) now rewards problem-solving literacy and application. Here are the key shifts: **1. Application Over Computation:** Instead of "Find HCF of 48, 60, 72," expect more word problems: "A gardener has rectangular plots and wants to divide them into identical square sections. What's the maximum side length?" This tests conceptual linking. **2. Divisibility as a Gateway:** Divisibility rules (especially for 11) are appearing earlier and more frequently in 3-mark questions, often paired with prime factorisation to verify answers. Examiners want to see you use the rule, not just compute. **3. Mixed Concepts in 5-Mark:** The new papers blend HCF/LCM with Sieve of Eratosthenes or prime factorisation in a single question. Example: "Prove that 323 is composite by writing its prime factorisation. Then find numbers that divide it using divisibility rules." This tests conceptual confidence. **4. Reduced Emphasis on Euclidean Algorithm Proof:** While HCF is still tested heavily, the formal Euclidean algorithm proof (showing steps exhaustively) is less critical now. Focus instead on using it as a verification tool. **5. Real-World Scenarios:** CBSE is pushing authentic contexts—sweets boxes, garden tiles, bell ringing, traffic light timing. Practice translating English-language word problems into mathematical models quickly. Study strategy: Balance 20% pure computation drills with 80% application and multi-concept blended questions.

Time management during an exam is as critical as knowing the content. Here's a battle-tested attempt order for Prime Time questions: **1-Minute Scan (First 60 seconds):** Read all questions on the paper. Mentally flag: (a) pure HCF/LCM (usually straightforward), (b) word problems (need careful setup), (c) divisibility rule questions (need precision), (d) Sieve/prime factorisation (may be conceptual). **Attempt Order:** • **Start with 1-mark questions (4–5 minutes total):** These are your confidence builders. Most are definitions or quick divisibility checks. Don't second-guess. If unsure, mark and move. • **Tackle 3-mark word problems next (12–15 minutes):** Allocate 3–4 minutes per question. Read once, underline key numbers, write prime factorisation or divisibility check, box your answer. Verify using the relationship HCF × LCM = product of numbers (if applicable). • **Attempt 5-mark questions (15–18 minutes):** These require two or three distinct steps. Write out prime factorisations first (even if not asked—it's a safety check). Then proceed step-by-step. Always show your work; partial credit is awarded for method. • **Final 2–3 minutes:** Revisit flagged 1-mark questions. Check one 3-mark arithmetic. Skip complex re-derivations. **Precision Tips:** • Always write prime factorisations in exponential form (2³ × 3², not 2 × 2 × 2 × 3 × 3). It's faster and examiners favour it. • Verify HCF/LCM answers by checking divisibility. Example: if you claim HCF(60, 90) = 30, confirm 60 ÷ 30 = 2 and 90 ÷ 30 = 3 (both integers). • For word problems, write one line stating what you're finding (e.g., "We need HCF to find maximum equal distribution") before solving. This earns method marks even if arithmetic fails. • Use the divisibility test for 11 immediately—it's a 20-second verification that often catches errors in factorisation. Start a 3-day free trial at cbsetutor.ai to access interactive drills on these strategies, live doubt-solving sessions, and adaptive question banks based on your weak spots.

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