# 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 Paper

**Explanation:**

Let’s analyze each statement step by step:

## 1. (p – r)(qs) is even.

– p is odd and r is even, so (p – r) is odd

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

– The product of an odd number and an even number is always even

Therefore, statement 1 is correct.

## 2. (q – s)qs is even.

– q is odd and s is even

– (q – s) is odd (odd minus even is odd)

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

– The product of an odd number and an even number is always even

Therefore, statement 2 is correct.

### 3. (q + r)²(p + s) is odd.

– q is odd and r is even, so (q + r) is odd

– (q + r)² is odd (odd squared is odd)

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

– The product of two odd numbers is odd

Therefore, statement 3 is correct.

Since all three statements are correct, the answer is: d) 1, 2 and 3