Calculate the HCF of `p^(3)q^(2)` and `p^(2)q,` provided that p and q ae prime numbers :

Options

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

Answer

To calculate the HCF (Highest Common Factor) of the expressions \( p^3q^2 \) and \( p^2q \), where \( p \) and \( q \) are prime numbers, we can follow these steps:

### Step 1: Write down the prime factorization
The first expression is \( p^3q^2 \), which can be broken down into its prime factors:
- \( p^3 = p \times p \times p \)
- \( q^2 = q \times q \)

Thus, \( p^3q^2 = p \times p \times p \times q \times q \).

The second expression is \( p^2q \):
- \( p^2 = p \times p \)
- \( q = q \)

So, \( p^2q = p \times p \times q \).

### Step 2: Identify the common factors
Now, we need to identify the common factors in both expressions:
- For \( p \): In \( p^3q^2 \), we have \( p \) appearing 3 times, and in \( p^2q \), we have \( p \) appearing 2 times. The minimum of these is \( p^2 \).
- For \( q \): In \( p^3q^2 \), we have \( q \) appearing 2 times, and in \( p^2q \), we have \( q \) appearing 1 time. The minimum of these is \( q^1 \) or simply \( q \).

### Step 3: Combine the common factors
Now, we combine the common factors to find the HCF:
- HCF = \( p^2 \times q \)

Thus, the HCF of \( p^3q^2 \) and \( p^2q \) is \( p^2q \).

### Final Answer
The HCF is \( p^2q \).

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