🔹 What is HCF (Highest Common Factor)?

HCF (Highest Common Factor) is the largest number that divides two or more numbers exactly without leaving a remainder.

🧠 HCF in Roman Urdu:

HCF woh sabse bara number hota hai jo do ya zyada numbers ko divide kar sakta hai bina kisi remainder ke.

📌 Example of HCF:

Find HCF of 12 and 18:

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18
• Common Factors: 1, 2, 3, 6
  ✅ HCF = 6

🔹 What is LCM (Least Common Multiple)?

LCM (Least Common Multiple) is the smallest number that is a multiple of two or more numbers.

🧠 LCM in Roman Urdu:

LCM woh sabse chhota number hota hai jo do ya zyada numbers ke tables mein common ho.

📌 Example of LCM:

Find LCM of 4 and 5:

• Multiples of 4: 4, 8, 12, 16, 20, 24, …
• Multiples of 5: 5, 10, 15, 20, 25, …
  ✅ LCM = 20

🔄 Key Differences Between HCF and LCM

Feature	HCF (Highest Common Factor)	LCM (Least Common Multiple)
Full Form	Largest number dividing all given numbers	Smallest multiple common to all numbers
Meaning	Focuses on common factors	Focuses on common multiples
Value	Always smaller than numbers	Always larger than numbers
Uses	Simplifying fractions, reducing ratios	Scheduling, finding common timings

📊 Example Summary (HCF vs LCM for 6 and 8)

• HCF (6, 8) = 2
• LCM (6, 8) = 24

✅ Real-Life Applications of HCF and LCM

🔹 Uses of HCF:

✔ Simplifying fractions (e.g., 12/18 simplifies to 2/3 using HCF 6)
✔ Dividing objects into equal groups

🔹 Uses of LCM:

✔ Finding common meeting times (e.g., two events repeating every 4 & 5 days)
✔ Scheduling tasks (e.g., two machines working at different intervals)
✔ Calculating repeating patterns (e.g., traffic light cycles)

LCM and HCF MCQs – Basic to Advanced Level PRACTICE MCQS WITH EXPLANATION

🔹 LCM & HCF – 5 Basic Level MCQs (Easy Practice Questions)

1. What is the HCF of 12 and 16?

A) 2
B) 4 ✅
C) 6
D) 8

📌 Explanation:

• Factors of 12 = 1, 2, 3, 4, 6, 12
• Factors of 16 = 1, 2, 4, 8, 16
• Common factors = 1, 2, 4 → HCF = 4

2. What is the LCM of 3 and 5?

A) 15 ✅
B) 8
C) 10
D) 30

📌 Explanation:

• Multiples of 3 = 3, 6, 9, 12, 15, …
• Multiples of 5 = 5, 10, 15, 20, …
• First common multiple = 15

3. The HCF of 7 and 13 is:

A) 1 ✅
B) 7
C) 13
D) 91

📌 Explanation:

• 7 and 13 are prime numbers → Only common factor = 1

4. LCM of 6 and 9 is:

A) 18 ✅
B) 36
C) 9
D) 12

📌 Explanation:

• Prime factors:
  • 6 = 2 × 3
  • 9 = 3 × 3
• LCM = 2 × 3 × 3 = 18

5. HCF of 15 and 25 is:

A) 5 ✅
B) 10
C) 15
D) 1

📌 Explanation:

• Common factors = 1, 5 → HCF = 5

🔴 LCM & HCF – 5 Advanced Level MCQs (Hard Questions)

6. What is the LCM of 8, 12, and 20?

A) 60
B) 80
C) 120 ✅
D) 240

📌 Explanation:

• Prime factorization:
  • 8 = 2³
  • 12 = 2² × 3
  • 20 = 2² × 5
• LCM = 2³ × 3 × 5 = 120

7. HCF of 72 and 120 is:

A) 6
B) 12
C) 24 ✅
D) 48

📌 Explanation:

• Factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
• Factors of 120 = 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
• Greatest common factor = 24

8. If LCM of 18 and 24 is 72, what is their HCF?

A) 6 ✅
B) 12
C) 8
D) 4

📌 Explanation:

• Formula: LCM × HCF = Product of numbers
• 72 × HCF = 18 × 24 → HCF = 432 ÷ 72 = 6

9. What is the HCF of 36, 60, and 72?

A) 6
B) 12 ✅
C) 18
D) 24

📌 Explanation:

• Common factors = 1, 2, 3, 4, 6, 12 → HCF = 12

10. Find the smallest number divisible by 8, 9, and 12:

A) 72
B) 96
C) 144 ✅
D) 360

📌 Explanation:

• LCM(8, 9, 12) = 2³ × 3² = 144

LCM & HCF Word Problems – MATH MCQs with Solutions

📌 LCM & HCF Word Problem MCQs

1. Rope Cutting Problem

Q: Two ropes of lengths 60 cm and 72 cm need to be cut into equal pieces. What is the maximum possible length of each piece?

Options:
A) 12 cm ✅
B) 15 cm
C) 18 cm
D) 24 cm

📌 Solution:

• This is an HCF (Highest Common Factor) problem since we need the largest equal length that divides both ropes.
• Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
• Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
• Common Factors: 1, 2, 3, 4, 6, 12
  ✔ Greatest possible length = HCF = 12 cm

Q: What is the smallest number divisible by 6, 8, and 9?

Options:

• A) 48
• B) 72 ✅
• C) 96
• D) 144

📌 Step-by-Step Solution

1. Understanding LCM

The LCM of multiple numbers is the smallest number that is a multiple of each of them. To find it, we use prime factorization.

2. Prime Factorization

First, break each number into its prime factors:

• 6 = 2 × 3
• 8 = 2 × 2 × 2 = 2³
• 9 = 3 × 3 = 3²

3. Calculating LCM

The LCM is found by taking the highest power of each prime factor present in the numbers:

• Highest power of 2: 2³ (from 8)
• Highest power of 3: 3² (from 9)

Now, multiply them together: LCM=23×32=8×9=72LCM=23×32=8×9=72

4. Verification

• 72 ÷ 6 = 12 (exact division)
• 72 ÷ 8 = 9 (exact division)
• 72 ÷ 9 = 8 (exact division)

Since 72 is divisible by all three numbers, it is the correct answer.

🔹 Why Not Other Options?

• A) 48 → Divisible by 6 and 8, but 48 ÷ 9 ≈ 5.33 (not exact) ❌
• C) 96 → Divisible by 6 and 8, but 96 ÷ 9 ≈ 10.66 (not exact) ❌
• D) 144 → Divisible by all, but not the smallest (72 is smaller) ❌

LCM and HCF Problem Solving: Find the Missing Number

Understanding the Relationship Between LCM and HCF

When working with two numbers, their Least Common Multiple (LCM) and Highest Common Factor (HCF) have a special mathematical relationship that can help us solve problems efficiently.

The Key Formula:

For any two numbers: LCM (a,b) × HCF (a,b) = a×b LCM (a,b)×HCF (a,b)=a×b

Q: Two numbers have LCM = 84 and HCF = 7. If one number is 21, find the other number.

Options:

• A) 28 ✅
• B) 35
• C) 42
• D) 56

📌 Step-by-Step Solution

1. Understand Given Values

• LCM of two numbers = 84
• HCF of two numbers = 7
• First number (a) = 21
• Second number (b) = ?

2. Apply the Fundamental Formula

Using the relationship:LCM(a,b)×HCF(a,b)=a×bLCM(a,b)×HCF(a,b)=a×b84×7=21×b84×7=21×b

3. Solve for the Unknown Number (b)

588=21×b588=21×bb=58821=28b=21588​=28

4. Verification

Let’s verify:

• HCF of 21 and 28:
  • Factors of 21: 1, 3, 7, 21
  • Factors of 28: 1, 2, 4, 7, 14, 28
  • Common factors: 1, 7 → HCF = 7 ✔
• LCM of 21 and 28:
  • Prime factors:
    • 21 = 3 × 7
    • 28 = 2² × 7
  • LCM = 2² × 3 × 7 = 4 × 3 × 7 = 84 ✔

🔹 Why Other Options Are Incorrect?

• B) 35 → LCM(21,35)=105 ≠ 84 ❌
• C) 42 → HCF(21,42)=21 ≠ 7 ❌
• D) 56 → LCM(21,56)=168 ≠ 84 ❌