Title: TNPSC Group IV – HCF and LCM Complete Short Notes: Rules, Fast Methods and Worked Examples
HCF and LCM are explicitly included in the TNPSC Group IV Aptitude syllabus. Following Day 01's lesson on percentages, today we study factors, multiples, prime numbers, HCF, LCM, calculation methods and common applications.
1. Factors and multiples A factor divides a number exactly, without a remainder. Factors of 18 are 1, 2, 3, 6, 9 and 18. A multiple is obtained by multiplying a number by 1, 2, 3, etc. Multiples of 6 include 6, 12, 18, 24 and 30.
2. Prime and composite numbers A prime number is greater than 1 and has exactly two positive factors: 1 and itself. Examples: 2, 3, 5, 7 and 11. A composite number has more than two positive factors. Examples: 4, 6, 8 and 9. The number 1 is neither prime nor composite. The number 2 is the only even prime number.
3. Quick divisibility rules 2: The last digit is 0, 2, 4, 6 or 8. 3: The sum of digits is divisible by 3. 4: The number formed by the last two digits is divisible by 4. 5: The last digit is 0 or 5. 6: The number is divisible by both 2 and 3. 8: The number formed by the last three digits is divisible by 8. 9: The sum of digits is divisible by 9. 10: The last digit is 0. 11: The difference between the sums of alternating digits is 0 or a multiple of 11.
4. HCF – Highest Common Factor (மீப்பெரு பொதுவகுத்தி) The HCF, also called the GCD, is the greatest positive integer that divides all given numbers exactly. Example: Common factors of 12 and 18 are 1, 2, 3 and 6. Therefore HCF = 6. Typical uses: Cutting ropes into the longest possible equal pieces, or dividing items into the largest identical groups without leftovers.
5. LCM – Least Common Multiple (மீச்சிறு பொது மடங்கு) The LCM is the smallest positive number that is a multiple of all the given numbers. Example: Multiples of 4 include 4, 8, 12…; multiples of 6 include 6, 12…; therefore LCM = 12. Typical uses: Finding when bells, lights or repeating events next occur together.
6. Prime factorisation method Write each number as a product of prime factors. For HCF, multiply the common prime factors using their lowest powers. For LCM, include every prime factor that appears, using its highest power.
Worked example 1 – HCF(48, 72) 48 = 2⁴ × 3; 72 = 2³ × 3². Take the lowest common powers: 2³ × 3 = 8 × 3. Answer: HCF = 24.
Worked example 2 – LCM(12, 18, 30) 12 = 2² × 3; 18 = 2 × 3²; 30 = 2 × 3 × 5. Take the highest powers: 2² × 3² × 5 = 4 × 9 × 5. Answer: LCM = 180.
7. Euclidean division method for HCF Divide the larger number by the smaller number and note the remainder. Repeat using the previous divisor and remainder. The last non-zero remainder is the HCF.
Worked example 3 – HCF(252, 198) 252 = 198 × 1 + 54 198 = 54 × 3 + 36 54 = 36 × 1 + 18 36 = 18 × 2 + 0 Answer: HCF = 18.
8. Important HCF–LCM identity For TWO positive integers only: First number × second number = HCF × LCM. Example: For 24 and 36, HCF = 12 and LCM = 72. 24 × 36 = 864 and 12 × 72 = 864. Do not directly apply this simple product identity to three or more numbers.
9. Special rules For co-prime numbers, HCF = 1 and LCM = their product. The HCF of two consecutive positive integers is 1. The HCF of two consecutive even positive integers is 2. If one number divides another exactly, HCF = the smaller number and LCM = the larger number. The HCF cannot exceed any of the given positive numbers; the LCM cannot be smaller than any of them.
10. Choosing HCF or LCM in word problems Clues such as “greatest equal length”, “largest identical size” and “divide without leftovers” usually indicate HCF. Clues such as “together again”, “shortest common time” and “smallest number divisible by all” usually indicate LCM.
Worked example 4 – Cutting ropes Two ropes are 84 metres and 126 metres long. Find the greatest equal length of pieces with no wastage and the total number of pieces. HCF(84, 126) = 42. Each piece is 42 metres. Number of pieces = 84/42 + 126/42 = 2 + 3 = 5. Answer: Five pieces, each 42 metres long.
Worked example 5 – Bells ringing together Three bells ring every 12, 18 and 30 seconds. If they ring together now, when will they next ring together? LCM(12, 18, 30) = 180 seconds = 3 minutes. Answer: After 3 minutes.
11. Common exam mistakes Do not take the highest powers for HCF and lowest powers for LCM; that reverses the rules. If numbers have no common factor other than 1, their HCF is 1, not 0. For “together again” questions, use LCM rather than HCF. The simple HCF × LCM product identity applies directly to two numbers, not to three numbers.
12. One-minute revision HCF = greatest common factor. LCM = smallest positive common multiple. HCF: Lowest powers of common prime factors. LCM: Highest powers of all prime factors. For two numbers: Product of numbers = HCF × LCM. Largest equal pieces → HCF. Repeating events together → LCM.
Textbook and book-back progress Relevant chapter identified: Tamil Nadu Samacheer Kalvi Class 6 Mathematics, Term 2, Chapter 1 “Numbers” (Factors and Multiples / HCF / LCM). Secondary textbook guides identify Exercise 1.2 as relevant. The complete original government textbook exercises and the 2026–27 edition have not yet been directly verified. Therefore, verified book-back items completed: 0; total item count: not yet verified. None of today's worked examples is labelled as a verbatim book-back question.
Today's HCF–LCM lesson is complete: definitions, rules, calculation methods and applications. Other Aptitude and Reasoning topics will follow on later days. The 7 PM practice set will include two questions on today's HCF–LCM topic.
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