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

To find the Highest Common Factor (HCF) of two numbers, we must identify the lowest power of all common prime factors in the given numbers.

Let's examine the given numbers:

• The prime factorization of the first number, \(2^2 \cdot 3^3\), is: \(2^2 \times 3^3\)
• The prime factorization of the second number, \(3^2 \cdot 2^3\), is: \(2^3 \times 3^2\)

Determine the common prime factors and their minimum powers:

• Common prime factor: \(2\), with the minimum power: \(2^2\)
• Common prime factor: \(3\), with the minimum power: \(3^2\)

The HCF is the product of these prime factors raised to their minimum powers:

\(HCF = 2^2 \times 3^2\)

Therefore, the HCF of \(2^2 \cdot 3^3\) and \(3^2 \cdot 2^3\) is \(2^2 \cdot 3^2\), which is Option \(2^2 \cdot 3^2\).