Question

The HCF of \(2^2 \cdot 3^3\) and \(3^2 \cdot 2^3\) is :

• A. 1
• B. \(2 \cdot 3\)
• C. \(2^2 \cdot 3^2\)
• D. \(2^3 \cdot 3^3\)

Hint:

To remember: HCF uses the Lowest power (Smallest), while LCM uses the Highest power.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The Highest Common Factor (HCF) of numbers expressed in prime factorization is the product of the smallest power of each common prime factor.
Step 2: Key Formula or Approach:
For two numbers \(A = p^a \cdot q^b\) and \(B = p^c \cdot q^d\):
\[ \text{HCF}(A, B) = p^{\min(a,c)} \cdot q^{\min(b,d)} \]
Step 3: Detailed Explanation:
1. Let \(A = 2^2 \cdot 3^3\) and \(B = 2^3 \cdot 3^2\).
2. Look at the prime factor 2: The powers are 2 and 3. The smaller power is \(2^2\).
3. Look at the prime factor 3: The powers are 3 and 2. The smaller power is \(3^2\).
4. Multiply these smallest powers: \(\text{HCF} = 2^2 \cdot 3^2\).
Step 4: Final Answer:
The HCF is \(2^2 \cdot 3^2\).