# Q. Let p, q, r and s be distinct positive integers.

Let p, q be odd and r, s be even. Consider the following statements:

1. (p – r) (qs) is even.

2. (q – s) q s is even.

3. (q + r)² (p + s) is odd.

Which of the statements given above are correct?

a) 1 and 2 only

b) 2 and 3 only

c) 1 and 3 only

d) 1, 2 and 3

Correct answer: d) 1, 2 and 3

#### Question from UPSC Prelims 2024 CSAT

**Explanation : **

Let’s analyze each statement given the conditions:

## Statement 1: (p – r)(qs) is even.

– (p – r): Since p is odd and r is even, p – r is odd.

– qs: q is odd and s is even, so qs is even.

– Therefore, (p – r)(qs) = odd × even = even.

– Conclusion: Statement 1 is correct.

### Statement 2: (q – s) q s is even.

– (q – s): q is odd and s is even, so q – s is odd.

– q × s: q is odd and s is even, making qs even.

– Thus, (q – s)qs = odd × even = even.

– Conclusion: Statement 2 is correct.

### Statement 3: (q + r)^2 (p + s) is odd.

– (q + r): q is odd and r is even, so q + r is odd. Squaring it, (q + r)^2, remains odd.

– (p + s): p is odd and s is even, so p + s is odd.

– Therefore, (q + r)^2 (p + s) = odd × odd = odd.

– Conclusion: Statement 3 is correct.

All three statements are correct.

Answer: d) 1, 2 and 3