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