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
36 = 2² × 3²
48 = 2⁴ × 3¹
Step 2: For HCF, take lowest powers of common primes
HCF = 2² × 3¹ = 4 × 3 = **12**
Step 3: For LCM, take highest powers of all primes
LCM = 2⁴ × 3² = 16 × 9 = **144**
Verification: 12 × 144 = 1728 = 36 × 48 ✓
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### Example 2: Division Method (for HCF) **Find HCF of 56 and 98 using division method.**
Step 1: Divide larger by smaller
98 ÷ 56 = 1, remainder = 42
Step 2: Divide previous divisor by remainder
56 ÷ 42 = 1, remainder = 14
Step 3: Continue until remainder = 0
42 ÷ 14 = 3, remainder = 0
**HCF = 14** (the last divisor)
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### Example 3: Word Problem (LCM type) **Three bells ring at intervals of 4, 6, and 9 minutes. If they ring together at 8:00 AM, when will they ring together again?**
Step 1: Find LCM of 4, 6, 9
4 = 2²
6 = 2 × 3
9 = 3²
LCM = 2² × 3² = 4 × 9 = 36 minutes
Step 2: Add to starting time
8:00 AM + 36 minutes = **8:36 AM**
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### Example 4: Word Problem (HCF type) **A room is 15 m long and 12 m wide. Find the largest square tile that can exactly cover the floor.**
Step 1: Recognise this as HCF problem (largest size that fits exactly)
HCF of 15 and 12
Step 2: 15 = 3 × 5, 12 = 2² × 3
HCF = 3
**Largest tile = 3 m × 3 m**
Number of tiles = (15 × 12) ÷ (3 × 3) = 180 ÷ 9 = 20 tiles
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### Example 5: Using Product Relationship **HCF of two numbers is 12 and their product is 1728. Find their LCM.**
Using: LCM × HCF = Product of numbers
LCM × 12 = 1728
LCM = 1728 ÷ 12 = **144**
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Common Mistakes
| Wrong Thinking | Correct Fix | |----------------|-------------| | Using LCM × HCF = product for three numbers | This formula works only for two numbers. For three or more, calculate LCM and HCF separately. | | Confusing when to use LCM vs HCF | LCM for "when together/simultaneously"; HCF for "largest possible/maximum that divides exactly." | | Taking highest powers for HCF | HCF uses lowest powers of common factors; LCM uses highest powers of all factors. | | Forgetting to include all prime factors in LCM | LCM must include every prime that appears in any number, with its highest power. | | Applying fraction formula incorrectly | For HCF of fractions: HCF of tops ÷ LCM of bottoms. Many students reverse this. |
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Quick Reference
**HCF = largest divisor common to all numbers**
**LCM = smallest number divisible by all given numbers**
**Two numbers: LCM × HCF = Product**
**Bells/events together → LCM**
**Largest tile/maximum piece → HCF**
**Co-prime → HCF = 1, LCM = product**
**HCF of fractions: HCF(num) / LCM(den)**
**LCM of fractions: LCM(num) / HCF(den)**
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You read the notes — now try one
Find the HCF of 48 and 72 using the prime factorisation method.
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