If two positive integers `m` and `n` are expressible in the form `m=p q^3` and `n=p^3q^2` , where `p ,\ q` are prime numbers, then HCF `(m ,\ n)=`

Options

A. `pq`
B. `pq^(2)`
C. `p^(3)q^(3)`
D. `p^(2)q^(2)`

Explanation

To find the HCF (Highest Common Factor) of the two positive integers \( m \) and \( n \) expressed as \( m = p q^3 \) and \( n = p^3 q^2 \), where \( p \) and \( q \) are prime numbers, we can follow these steps:

### Step 1: Write down the expressions for \( m \) and \( n \)
We have:
- \( m = p q^3 \)
- \( n = p^3 q^2 \)

### Step 2: Factorize \( m \) and \( n \)
From the expressions, we can see:
- For \( m \): The prime factorization is \( p^1 \cdot q^3 \)
- For \( n \): The prime factorization is \( p^3 \cdot q^2 \)

### Step 3: Identify the common prime factors
The common prime factors between \( m \) and \( n \) are \( p \) and \( q \).

### Step 4: Determine the minimum power of each common prime factor
- For \( p \): The minimum power is \( \min(1, 3) = 1 \)
- For \( q \): The minimum power is \( \min(3, 2) = 2 \)

### Step 5: Write the HCF using the common prime factors
The HCF is obtained by multiplying the common prime factors raised to their minimum powers:
\[ \text{HCF}(m, n) = p^{\min(1, 3)} \cdot q^{\min(3, 2)} = p^1 \cdot q^2 = p q^2 \]

### Conclusion
Thus, the HCF of \( m \) and \( n \) is:
\[ \text{HCF}(m, n) = p q^2 \]