IF two positive integers a and b are written as `a=x^3y^2 and b=xy^3` where x and y are prime numbers, then the HCF (a,b) is

Options

A. xy
B. `xy^2`
C. `x^3y^3`
D. `x^2y^2`

Answer

To find the HCF (Highest Common Factor) of the two positive integers \( a \) and \( b \) given as \( a = x^3y^2 \) and \( b = xy^3 \), where \( x \) and \( y \) are prime numbers, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the prime factorization of \( a \) and \( b \)**:
- \( a = x^3y^2 \) means \( a \) has \( x \) raised to the power of 3 and \( y \) raised to the power of 2.
- \( b = xy^3 \) means \( b \) has \( x \) raised to the power of 1 and \( y \) raised to the power of 3.

2. **Write down the prime factors**:
- For \( a \): The prime factors are \( x, x, x, y, y \) (3 times \( x \) and 2 times \( y \)).
- For \( b \): The prime factors are \( x, y, y, y \) (1 time \( x \) and 3 times \( y \)).

3. **Find the HCF by taking the lowest power of each prime factor**:
- For the prime \( x \):
- In \( a \), the power is 3.
- In \( b \), the power is 1.
- The lowest power is \( 1 \), so we take \( x^1 \).
- For the prime \( y \):
- In \( a \), the power is 2.
- In \( b \), the power is 3.
- The lowest power is \( 2 \), so we take \( y^2 \).

4. **Combine the results**:
- Therefore, the HCF of \( a \) and \( b \) is:
\[ \text{HCF}(a, b) = x^1y^2 = xy^2 \]

### Final Answer:
The HCF of \( a \) and \( b \) is \( xy^2 \).