Title: HCF and LCM – Complete Notes, Shortcuts and Seven Fully Worked Examples
1. Factors, multiples and prime numbers A factor divides a number exactly, leaving no remainder. Factors of 12 are 1, 2, 3, 4, 6 and 12. A multiple is obtained by multiplying a number by 1, 2, 3, etc. Multiples of 6 are 6, 12, 18, 24, etc. A prime number is a natural number greater than 1 with exactly two positive factors: 1 and itself. The number 2 is the only even prime. The number 1 is neither prime nor composite. Prime factorisation expresses a number as a product of primes. For example, 60 = 2² × 3 × 5.
2. Highest Common Factor (HCF / மீப்பெரு பொதுக்காரணி) The HCF is the largest positive integer that divides all the given numbers without a remainder. For 12 and 18, the common factors are 1, 2, 3 and 6. Therefore HCF(12,18) = 6. Using prime factorisation, select only the prime factors common to every number, taking the LOWEST power of each.
3. Least Common Multiple (LCM / மீச்சிறு பொது மடங்கு) The LCM is the smallest positive integer divisible by every given number. Multiples of 4 are 4, 8, 12, 16, etc. Multiples of 6 are 6, 12, 18, etc. Therefore LCM(4,6) = 12. Using prime factorisation, include every prime factor that appears, taking the HIGHEST power of each.
4. Quick divisibility tests 2: last digit even. 3: digit sum divisible by 3. 4: last two digits divisible by 4. 5: last digit 0 or 5. 6: divisible by both 2 and 3. 8: last three digits divisible by 8. 9: digit sum divisible by 9. 10: last digit 0. 11: difference between alternating-position digit sums is zero or a multiple of 11.
5. Important formulas and shortcuts For TWO positive integers only: First number × Second number = HCF × LCM. If one number divides the other exactly, HCF = smaller number and LCM = larger number. Two numbers are co-prime if their HCF is 1; their LCM then equals their product. Consecutive natural numbers have HCF 1. Consecutive even numbers have HCF 2. The HCF cannot exceed the smallest given number. The LCM cannot be less than the largest given number.
6. Seven fully worked examples Example 1 – HCF of 84 and 126: 84 = 2² × 3 × 7; 126 = 2 × 3² × 7. Use the lowest powers of common primes: 2 × 3 × 7 = 42. Answer: HCF = 42.
Example 2 – LCM of 18, 24 and 30: 18 = 2 × 3²; 24 = 2³ × 3; 30 = 2 × 3 × 5. Use the highest powers: 2³ × 3² × 5 = 8 × 9 × 5 = 360. Answer: LCM = 360.
Example 3 – The HCF of two numbers is 12 and their LCM is 180. One number is 36. Find the other. 36 × Other number = 12 × 180 = 2160. Other number = 2160 ÷ 36 = 60. Check: HCF(36,60)=12 and LCM(36,60)=180. Answer: 60.
Example 4 – Pack 48 pens and 72 pencils into the greatest possible number of identical packs without leftovers. How many packs? HCF(48,72)=24. Each pack contains 48÷24=2 pens and 72÷24=3 pencils. Answer: 24 packs.
Example 5 – Three bells ring every 12, 18 and 30 seconds. If they ring together now, when will they next ring together? 12 = 2² × 3; 18 = 2 × 3²; 30 = 2 × 3 × 5. LCM = 2² × 3² × 5 = 180 seconds = 3 minutes. Answer: After 3 minutes.
Example 6 – Find the greatest four-digit number exactly divisible by 12, 18 and 30. LCM(12,18,30)=180. The greatest whole multiple of 180 not exceeding 9999 is 180 × 55. Answer: 9900.
Example 7 – Find the smallest integer greater than 15 that leaves remainder 3 when divided by each of 8, 12 and 15. LCM(8,12,15)=120. A number of the required form is 120k + 3. For k=1, 120 + 3 = 123, which is greater than 15. Check: 123 leaves remainder 3 on division by 8, 12 or 15. Answer: 123.
7. When should you use HCF or LCM? Usually use HCF for the greatest number of identical packs, the largest equal measuring vessel, or the longest equal pieces with no remainder. Usually use LCM for the next time repeating events happen together or the smallest number divisible by several given numbers. Always check the full conditions rather than relying only on keywords.
8. Common mistakes Use LOWEST prime powers for HCF and HIGHEST for LCM. Do not apply the two-number product formula directly to three numbers. One is not prime. Co-prime numbers need not individually be prime: 8 and 9 are co-prime.
9. One-minute revision HCF = largest common factor; LCM = smallest common multiple. For 12 and 18: HCF = 6; LCM = 36. For two positive integers: product = HCF × LCM. Consecutive natural numbers have HCF 1. Repeated events meeting again usually need LCM; maximum equal divisions usually need HCF.
10. Textbook and book-back tracking Reference: Tamil Nadu Samacheer Kalvi Class 6 Mathematics, Term 2, Unit 1 – Numbers, especially Exercise 1.2 (HCF and LCM). A publicly available transcription labelled as the 2025 edition identifies Exercise 1.2 main questions 1–13, with subquestions, and further mixed practice in Exercise 1.3. However, the complete original government PDF and 2026–27 edition have not yet been independently checked: source not yet verified. The seven examples in this article are ORIGINAL WORKED EXAMPLES, not verified textbook book-back questions or TNPSC previous-year questions. Verified book-back questions completed: 0. The full exercise-item count remains unconfirmed. Once the original exercises are checked, all syllabus-relevant questions will be covered through separately numbered 7 PM practice batches.
11. Revision schedule Revise this topic on 18, 20, 24 and 31 October 2026 (1/3/7/14 days). Briefly revise yesterday's Ratio and Proportion lesson today. Tonight's 7 PM practice will contain 15 headline MCQs: 8 General Tamil, 5 General Studies and 2 Aptitude.
