If p,q,r be integers such that `p^2 = q^2.r` then :

Options

A. a. p is an even number
B. b. q is an even number
C. c. r is an even number
D. d. r is a perfect square

Answer

To solve the equation \( p^2 = q^2 \cdot r \) where \( p, q, r \) are integers, we will analyze the parity (evenness or oddness) of the integers involved.

### Step-by-Step Solution:

1. **Understanding the Equation**:
We start with the equation \( p^2 = q^2 \cdot r \). This means that the square of \( p \) is equal to the product of the square of \( q \) and \( r \).

**Hint**: Recognize that \( p^2 \) is always non-negative, and analyze the implications of the equation.

2. **Analyzing the Parity of \( p \)**:
- If \( p \) is **even**, then \( p^2 \) is also **even**.
- Therefore, \( q^2 \cdot r \) must also be **even**.

**Hint**: Recall that the product of two integers is even if at least one of them is even.

3. **Considering \( q \)**:
- If \( q \) is **even**, then \( q^2 \) is **even**. Thus, \( r \) can be either even or odd.
- If \( q \) is **odd**, then \( q^2 \) is **odd**. For \( q^2 \cdot r \) to be even, \( r \) must be **even**.

**Hint**: Analyze the implications of \( q \) being even or odd on \( r \).

4. **Analyzing the Parity of \( r \)**:
- If \( r \) is **even**, then \( q^2 \cdot r \) is even regardless of whether \( q \) is even or odd.
- If \( r \) is **odd**, then \( q^2 \) must be even, which implies \( q \) must be even.

**Hint**: Determine how the parity of \( r \) affects the possible values of \( q \).

5. **Conclusion**:
- If \( p \) is even, then both \( q \) and \( r \) can be even.
- If \( p \) is odd, then \( q \) must be odd and \( r \) must also be odd.

Therefore, we conclude that:
- \( p, q, r \) can all be even, or
- \( p, q, r \) can all be odd.

### Final Result:
The integers \( p, q, r \) must either all be even or all be odd.