LCM and HCF
Study Notes for MP TET
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Overview
LCM (Lowest Common Multiple) and HCF (Highest Common Factor) form the backbone of number-system problems in MP TET. These concepts connect directly to fractions, divisibility, ratio-proportion, and time-work problems. Questions appear both as direct calculations and as word problems involving bells ringing together, circular tracks, distribution of items, and measurement scenarios.
For MP TET, you must be fluent in three methods — prime factorisation, division method, and the product relationship formula. Expect 2–4 questions combining LCM/HCF with real-life contexts. Mastering this topic also strengthens your ability to teach these concepts to primary and upper-primary students using concrete examples.
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Key Concepts
- **HCF (Highest Common Factor)**: The largest number that divides two or more numbers exactly. Also called GCD (Greatest Common Divisor).
- **LCM (Lowest Common Multiple)**: The smallest number that is exactly divisible by two or more numbers.
- **Co-prime numbers**: Two numbers are co-prime if their HCF = 1 (e.g., 8 and 15).
- **Product relationship**: For any two numbers a and b, LCM × HCF = a × b. This works only for two numbers, not three or more.
- **HCF of fractions**: HCF of numerators ÷ LCM of denominators.
- **LCM of fractions**: LCM of numerators ÷ HCF of denominators.
- **Key insight**: HCF ≤ smaller number ≤ larger number ≤ LCM. The HCF always divides the LCM.
- **Word problem signals**: "Together/simultaneously" usually means LCM; "largest possible" or "maximum size" usually means HCF.
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Formulas / Key Facts
| Formula/Fact | Context | |--------------|---------| | LCM × HCF = a × b | Valid only for two numbers a and b | | HCF(fractions) = HCF of numerators / LCM of denominators | Finding HCF of fractions | | LCM(fractions) = LCM of numerators / HCF of denominators | Finding LCM of fractions | | HCF of co-primes = 1 | Definition of co-prime numbers | | LCM of co-primes = product of the numbers | Direct multiplication | | If a divides b, then HCF(a,b) = a and LCM(a,b) = b | When one number is a factor of another | | For three numbers: LCM × HCF ≠ product | Product rule fails for more than two numbers |
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Worked Examples
### Example 1: Prime Factorisation Method **Find HCF and LCM of 36 and 48.**
Step 1: Prime factorise both numbers