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

Options

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

Answer

To find the HCF (Highest Common Factor) of the two positive integers \( a \) and \( b \) given as:

- \( a = x^3 y^2 \)
- \( b = x y^3 \)

where \( x \) and \( y \) are prime numbers, we can follow these steps:

### Step 1: Write down the prime factorization of \( a \) and \( b \)

- For \( a \):
\[ a = x^3 y^2 = x \cdot x \cdot x \cdot y \cdot y \]

- For \( b \):
\[ b = x y^3 = x \cdot y \cdot y \cdot y \]

### Step 2: Identify the common prime factors

Now, we will identify the common prime factors in \( a \) and \( b \):

- The prime factor \( x \) appears in both \( a \) and \( b \).
- The prime factor \( y \) appears in both \( a \) and \( b \).

### Step 3: Determine the lowest power of each common prime factor

- For the prime factor \( x \):
- In \( a \), the power of \( x \) is \( 3 \).
- In \( b \), the power of \( x \) is \( 1 \).
- The minimum power is \( \min(3, 1) = 1 \).

- For the prime factor \( y \):
- In \( a \), the power of \( y \) is \( 2 \).
- In \( b \), the power of \( y \) is \( 3 \).
- The minimum power is \( \min(2, 3) = 2 \).

### Step 4: Write the HCF using the common prime factors and their lowest powers

Now, we can write the HCF as:
\[ \text{HCF}(a, b) = x^{\min(3, 1)} \cdot y^{\min(2, 3)} = x^1 \cdot y^2 = xy^2 \]

### Final Answer

Thus, the HCF of \( a \) and \( b \) is:
\[ \text{HCF}(a, b) = xy^2 \]

---