Three of the five positive integers p, q, r, s, t are even and two of them are odd (not necessarily in…
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?
1 only
2 only
Both 1 and 2
Neither 1 nor 2
Solution
Statement 1: The sum/difference $p+q+r-s-t$ has parity equal to the number of odd terms among p, q, r, s, t. There are exactly 2 odd integers, so the count of odd terms is 2 (even), making the expression definitely even. Statement 1 is correct. Statement 2: $2p + q + 2r - 2s + t$ has parity equal to $q + t$ (since $2p$, $2r$, $2s$ are even). The parity of $q + t$ depends on which of the five are odd, so it is not definitely odd. Statement 2 is not correct. Only statement 1 is correct.