# Q. 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

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

Correct Answer: (a) 1 only

#### Question from UPSC Prelims 2023 CSAT

**Explanation : **

## Odd-even integers p, q, r, s, t

The sum or difference of two even numbers is always even, and the sum or difference of two odd numbers is always even. So, the sum or difference of any number of even numbers is always even.

**For statement 1,** since p, q, r, s, t are five integers where three are even and two are odd, we can group the even and odd numbers separately. The sum of three even numbers minus the sum of two odd numbers is even – even, which is definitely even.

**For statement 2**, the expression 2p + q + 2r – 2s + t can be rearranged as 2(p + r – s) + (q + t). The term 2(p + r – s) is definitely even as it is a product of 2 and some integer. However, (q + t) is the sum of two integers where one is even and the other is odd, which is definitely odd. Therefore, the sum of an even number and an odd number is odd. So, statement 2 is not definitely odd, it depends on the values of q and t.

Therefore, only statement 1 is correct.