Step 1: Understanding the Concept:
HCF stands for Highest Common Factor. It is the largest positive integer that divides each of the given numbers without leaving a remainder.
It is also known as the Greatest Common Divisor (GCD).
Finding the HCF is vital in simplifying fractions and solving problems related to partitioning objects into equal sizes.
Key Formula or Approach:
There are multiple methods to find the HCF:
1. Factor Listing Method: Listing all factors and finding the largest common one.
2. Prime Factorization Method: Breaking numbers into prime factors and taking the product of the lowest powers of common primes.
3. Division Method: Dividing the larger number by the smaller and continuing with the remainder.
Step 2: Detailed Explanation:
Method 1: Prime Factorization
Let's find the prime factors of 24:
\[ 24 = 2 \times 12 \]
\[ 24 = 2 \times 2 \times 6 \]
\[ 24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3^1 \]
Now, let's find the prime factors of 36:
\[ 36 = 2 \times 18 \]
\[ 36 = 2 \times 2 \times 9 \]
\[ 36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2 \]
To find the HCF, identify the common prime bases and take their lowest powers found in either factorization:
Common primes are 2 and 3.
Lowest power of 2 is \( 2^2 \).
Lowest power of 3 is \( 3^1 \).
\[ HCF = 2^2 \times 3^1 = 4 \times 3 = 12 \]
Method 2: Listing Factors
Factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors are: 1, 2, 3, 4, 6, 12
The highest among these common factors is 12.
Step 3: Final Answer:
The Highest Common Factor of 24 and 36 is 12.
This matches Option (B).