LCM and HCF
Overview
LCM (Lowest Common Multiple) and HCF (Highest Common Factor) form the backbone of number theory at the primary level. These concepts appear directly in JTET Paper I mathematics questions and also underpin problems on fractions, ratio-proportion, time-and-work, and word problems involving grouping or distribution.
For the exam, you must be able to find LCM and HCF using multiple methods (listing, prime factorisation, division), understand when to apply each concept in word problems, and know the key relationship that connects them. Mastery here also helps in simplifying fractions and solving problems on bells ringing together, circular tracks, and equal distribution.
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Key Concepts
- **Factor**: A number that divides another number exactly (without remainder). Factors of 12 are 1, 2, 3, 4, 6, 12.
- **Multiple**: A number obtained by multiplying a given number by any whole number. Multiples of 4 are 4, 8, 12, 16, 20...
- **HCF (Highest Common Factor)**: The greatest number that divides two or more numbers exactly. Also called GCD (Greatest Common Divisor).
- **LCM (Lowest Common Multiple)**: The smallest number that is a multiple of two or more numbers.
- **Co-prime numbers**: Two numbers whose HCF is 1 (e.g., 8 and 15).
- **Key relationship**: For any two numbers a and b, LCM × HCF = a × b. This is exam-critical.
- **When to use HCF**: Problems involving division, distribution into equal groups, cutting into equal pieces, finding the largest tile size.
- **When to use LCM**: Problems involving common time, bells ringing together, circular track meetings, finding common multiples.
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Formulas / Key Facts
| Formula / Fact | Context | |----------------|---------| | LCM × HCF = Product of two numbers | Works only for two numbers; very useful for finding one when the other is known | | HCF of co-prime numbers = 1 | Numbers like 9 and 16, or any two consecutive numbers | | LCM of co-prime numbers = their product | Since HCF = 1, LCM = a × b | | HCF(a, b) ≤ smaller number | HCF can never exceed the smallest of the given numbers | | LCM(a, b) ≥ larger number | LCM is at least as large as the biggest number | | HCF of numbers divides their LCM | Always true; useful for verification | | If a divides b, then HCF = a and LCM = b | Example: HCF(4, 12) = 4, LCM(4, 12) = 12 |
**Methods to find HCF:** 1. Listing factors method 2. Prime factorisation — take common prime factors with lowest powers 3. Continued division (Euclid's method)