Three of the five positive integers p, q, r, s, t are even and two of them are odd (not necessarily in order). Consider the following : 1. p + q + r - s - t is definitely even. 2. 2p + q + 2r - 2s + t is definitely odd. Which of the above statements is/are correct ?

(a) 1 only ✓ UPSC's answer
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

Why the answer is (a)

• There are exactly three even and two odd integers among p, q, r, s, t.

• In statement 1, subtraction does not change parity, so p + q + r - s - t has the same parity as p + q + r + s + t.

• The sum of three even numbers and two odd numbers is even, so statement 1 is definitely even.

• In statement 2, 2p, 2r and 2s are even, so the parity depends only on q + t.

• q and t are not specified as odd, so q + t can be even or odd; statement 2 is not definitely odd.

• Hence only statement 1 is correct, so option (a) follows.

Why the other options are wrong

(b) 2 only: Statement 1 is definitely even because the total parity of three even and two odd integers is even.
(c) Both 1 and 2: Statement 2 is not definitely odd because q and t may both be even or both odd, making q + t even.
(d) Neither 1 nor 2: Statement 1 is definitely even, so the option saying neither statement is correct is wrong.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.