Question

If \( x, y \) and \( z \) are distinct prime numbers, then the H.C.F. of \( x^2 y^3 z \) and \( x^3 y z^2 \) is:

• A. \( x^2 y z \)
• B. \( x y z^2 \)
• C. \( x^3 y^3 z^3 \)
• D. \( x^2 y^2 z^2 \)

Hint:

To find the HCF of two expressions, take the lowest power of each common factor.

Answer 1

The Correct Option is A

We are given two expressions: \[ x^2 y^3 z \quad \text{and} \quad x^3 y z^2. \] Step 1: The highest common factor (HCF) is found by taking the lowest power of each common factor in both expressions. Since \( x, y, z \) are distinct prime numbers, we examine the powers of each factor in both expressions. - For \( x \), the powers are \( 2 \) and \( 3 \), so the lowest power is \( x^2 \). - For \( y \), the powers are \( 3 \) and \( 1 \), so the lowest power is \( y \). - For \( z \), the powers are \( 1 \) and \( 2 \), so the lowest power is \( z \). Step 2: Therefore, the HCF is: \[ \text{HCF} = x^2 y z. \] Thus, the HCF of \( x^2 y^3 z \) and \( x^3 y z^2 \) is \( x^2 y z \).