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?

To solve this, we look at the properties of even and odd numbers.

Statement 1: p + q + r - s - t The sum or difference of integers follows parity rules. Whether you add or subtract five numbers, the result's parity depends on the number of odd integers in the set. Here, we have exactly two odd numbers. When you combine two odd numbers (either by addition or subtraction), the result is always even. Adding three even numbers to that result will keep it even. Therefore, the expression is always even, regardless of which specific variables are odd. Statement 1 is correct.

Statement 2: 2p + q + 2r - 2s + t We can simplify this by looking at the coefficients. 2p, 2r, and 2s are always even because any integer multiplied by 2 results in an even number. The expression essentially behaves like: Even + q + Even - Even + t. This simplifies to: Even + q + t. The parity of the total depends entirely on (q + t). If q and t are the two odd numbers, their sum is even. If one is odd and one is even, their sum is odd. If both are even, their sum is even. Since we do not know which variables are odd, the result is not definitely odd. Statement 2 is incorrect.

Conclusion: Only Statement 1 is correct. The correct option is A.