Fractions

1. Understanding Factors and Prime Factors

What is a Factor?

A factor is a number that divides another number exactly without leaving a remainder.

Example: Find the factors of 24, 72, and 100.

• 24 = 2 × 2 × 2 × 3
• 72 = 2 × 2 × 2 × 3 × 3
• 100 = 2 × 2 × 5 × 5

🔹 Key Insight: The numbers 2, 3, and 5 are the prime factors here because they cannot be broken down further.

What is a Prime Factor?

A prime factor is a factor that is also a prime number (divisible only by 1 and itself).

Example: Factorize 57.

• 57 = 3 × 19
  🔹 Explanation: Both 3 and 19 are prime numbers, making them prime factors of 57.

2. Highest Common Factor (HCF)

Definition:

The HCF of two or more numbers is the largest number that divides all of them exactly.

Problem: Find the HCF of 24, 72, and 100.

Step-by-Step Solution:

1. Factorize each number:
   • 24 = 2 × 2 × 2 × 3
   • 72 = 2 × 2 × 2 × 3 × 3
   • 100 = 2 × 2 × 5 × 5
2. Identify common prime factors:
   • The common factors in all three numbers are two 2’s (i.e., 2 × 2).
3. Multiply the common factors:
   • HCF = 2 × 2 = 4

🔹 Final Answer: The HCF of 24, 72, and 100 is 4.

3. Lowest Common Multiple (LCM)

Definition:

The LCM of two or more numbers is the smallest number that is a multiple of all of them.

Problem: Find the LCM of 84, 92, and 76.

Step-by-Step Solution: Divide all numbers by the smallest prime (2, 3, 5, etc.) until no further division is possible:

2 | 84, 92, 76
2 | 42, 46, 38
3 | 21, 23, 19

1. Stop when no common factor remains:
   • The remaining numbers are 7, 23, and 19 (all primes).
2. Multiply all divisors and remaining primes:
   • LCM = 2 × 2 × 3 × 7 × 23 × 19 = 36,708

🔹 Final Answer: The LCM of 84, 92, and 76 is 36,708.

4. Practical Applications of HCF and LCM

Why are HCF and LCM Important?

• HCF is used to simplify fractions.
• LCM is crucial for adding/subtracting fractions and solving time-based problems.

Example Problem:
Find the LCM of 36, 108, and 60.

Solution:Divide by smallest primes:

2 | 36, 108, 60
2 | 18, 54, 30
3 | 9, 27, 15
3 | 3, 9, 5

Multiply all divisors and remaining numbers:

• LCM = 2 × 2 × 3 × 3 × 3 × 5 = 540

🔹 Final Answer: The LCM of 36, 108, and 60 is 540.

5. Quick Summary Table

6. Key Takeaways

✅ Factors break numbers into smaller multipliers.
✅ HCF simplifies large calculations.
✅ LCM helps in fraction operations and scheduling problems.

🔹 Final Tip: Always start factorization with the smallest prime (2) and proceed upwards (3, 5, 7…).