LCM and HCF — Study Notes for RRB NTPC

Overview

LCM (Least Common Multiple) and HCF (Highest Common Factor, also called GCD) form the foundation for solving numerous problems in RRB NTPC Mathematics. These concepts appear directly in 2–4 questions and indirectly support topics like fractions, time-and-work cycles, and recurring events.

The exam tests three main skills: (1) finding LCM/HCF of whole numbers using prime factorization or division methods, (2) applying these to fractions and decimals, and (3) solving word problems where you must recognize whether the situation demands LCM (finding a common occurrence or total quantity) or HCF (finding maximum equal parts or greatest measure). Mastery requires understanding the relationship between LCM and HCF, especially the formula **LCM × HCF = Product of two numbers**, which frequently shortcuts calculation in MCQs.

Most mistakes arise from confusing when to use LCM versus HCF in word problems. Practice identifying keywords: "simultaneously," "together again," "least" suggests LCM; "maximum," "greatest," "largest possible" suggests HCF. With 90–120 seconds per question, you must execute prime factorization quickly and apply formulas confidently.

Key Concepts

• **HCF (Highest Common Factor)** is the largest number that divides all given numbers exactly. Think of it as the greatest measure that fits into each quantity without remainder.

• **LCM (Least Common Multiple)** is the smallest number that is a multiple of all given numbers. It represents the first point where all cycles align or the smallest quantity divisible by each number.

• **Prime factorization method**: Express each number as a product of primes. HCF = product of common prime factors with lowest powers; LCM = product of all prime factors with highest powers.

• **Division method (for HCF)**: Repeatedly divide the larger number by the smaller and replace the larger with the remainder until remainder is zero. The last divisor is the HCF. Also called Euclidean algorithm.

• **Relationship formula**: For any two numbers a and b, **LCM(a, b) × HCF(a, b) = a × b**. This shortcut is gold for RRB NTPC MCQs when three of the four values are known.

• **LCM/HCF of fractions**: **HCF of fractions = HCF of numerators / LCM of denominators**; **LCM of fractions = LCM of numerators / HCF of denominators**. Apply the standard methods to tops and bottoms separately.

• **LCM/HCF of decimals**: Convert decimals to like decimals (same number of decimal places), remove the decimal to treat as whole numbers, find LCM/HCF, then reinsert the decimal at the same place.

• **Co-prime numbers**: Two numbers with HCF = 1 (no common factor except 1). For co-primes, LCM = product of the numbers.

Formulas / Key Facts

1. **LCM × HCF = Product of two numbers** (a × b) — Use when any three quantities are known to find the fourth. 2. **HCF by prime factorization** = Product of common primes raised to the smallest power present in all numbers. 3. **LCM by prime factorization** = Product of all primes raised to the highest power present in any number. 4. **HCF of fractions** = (HCF of numerators) / (LCM of denominators). 5. **LCM of fractions** = (LCM of numerators) / (HCF of denominators). 6. **For decimals**: Equalize decimal places, find LCM/HCF of resulting integers, place decimal point back at the original position. 7. **LCM of three numbers a, b, c**: First find LCM(a, b), then find LCM(result, c). Similarly for HCF. 8. **If HCF of a and b is h**, then a and b can be written as a = hm and b = hn where m and n are co-prime.

Worked Examples

**Example 1: Find HCF and LCM of 72 and 120**

*Step 1:* Prime factorize each number.

• 72 = 2³ × 3²
• 120 = 2³ × 3¹ × 5¹

*Step 2:* HCF = product of common primes with lowest powers = 2³ × 3¹ = 8 × 3 = **24**

*Step 3:* LCM = product of all primes with highest powers = 2³ × 3² × 5¹ = 8 × 9 × 5 = **360**

*Verification:* LCM × HCF = 360 × 24 = 8640; Product = 72 × 120 = 8640 ✓

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**Example 2: Find LCM and HCF of 2/3, 4/9, and 5/6**

*Step 1:* Identify numerators (2, 4, 5) and denominators (3, 9, 6).

*Step 2:* LCM of numerators = LCM(2, 4, 5) = 20

*Step 3:* HCF of numerators = HCF(2, 4, 5) = 1

*Step 4:* LCM of denominators = LCM(3, 9, 6) = 18

*Step 5:* HCF of denominators = HCF(3, 9, 6) = 3

*Step 6:* **HCF of fractions** = 1/18; **LCM of fractions** = 20/3

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**Example 3: Three bells ring at intervals of 12, 15, and 20 minutes. If they ring together at 8:00 AM, when will they ring together again?**

*Recognize:* "Together again" = all cycles align = **LCM problem**

*Step 1:* Find LCM(12, 15, 20).

• 12 = 2² × 3
• 15 = 3 × 5
• 20 = 2² × 5
• LCM = 2² × 3 × 5 = 60 minutes

*Step 2:* They ring together every 60 minutes = 1 hour.

*Answer:* Next simultaneous ring at **9:00 AM**.

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**Example 4: Find the greatest number that divides 285 and 1249, leaving remainders 9 and 7 respectively.**

*Recognize:* "Greatest number that divides leaving remainders" = **HCF problem** after removing remainders.

*Step 1:* Subtract remainders: 285 – 9 = 276; 1249 – 7 = 1242

*Step 2:* Find HCF(276, 1242).

• 276 = 2² × 3 × 23
• 1242 = 2 × 3³ × 23
• HCF = 2 × 3 × 23 = **138**

*Answer:* The greatest number is **138**.

Common Mistakes

**Mistake 1:** Confusing LCM and HCF in word problems. "Least time to meet again" requires LCM, but students use HCF. **Fix:** Keywords matter — "together/simultaneously/least" → LCM; "maximum/greatest/largest part" → HCF.

**Mistake 2:** Applying the formula LCM × HCF = a × b to three or more numbers. **Fix:** This formula works **only for two numbers**. For three numbers, compute step-by-step or use prime factorization.

**Mistake 3:** In fraction LCM/HCF, swapping numerator and denominator operations. **Fix:** Memorize: HCF of fractions = (HCF top) / (LCM bottom); LCM of fractions = (LCM top) / (HCF bottom). Write it down until automatic.

**Mistake 4:** Forgetting to reinsert the decimal point when finding LCM/HCF of decimals. **Fix:** Count decimal places in the original numbers. After calculating with integers, place the decimal point at the same total count from the right.

**Mistake 5:** Assuming HCF is always smaller than the numbers — fails when one number divides another. **Fix:** If a divides b, then HCF(a, b) = a. For example, HCF(12, 36) = 12, not something smaller.

Quick Reference

• **LCM × HCF = a × b** (two numbers only) — fast shortcut for MCQs.
• **HCF of fractions = (HCF of tops) / (LCM of bottoms)**
• **LCM of fractions = (LCM of tops) / (HCF of bottoms)**
• **Word problem clues:** "together again/simultaneously/least" → LCM; "maximum equal parts/greatest measure" → HCF.
• **Decimals:** Make equal decimal places → compute as integers → restore decimal point.
• **Co-prime means HCF = 1**, so LCM = product of the numbers.