Correct Option
Statement 1: p + q + r – s – t is definitely even.
Among the five positive integers p, q, r, s, t, there are three even (E) and two odd (O) numbers. The parity of a sum or difference of integers is determined by the number of odd terms involved. An even count of odd terms results in an even sum/difference, while an odd count of odd terms results in an odd sum/difference.
In the expression p + q + r – s – t, the operations of addition and subtraction do not alter the individual parity of the terms. Therefore, the parity of the entire expression is determined by the combined parity of the five integers. Since there are three even numbers and two odd numbers among p, q, r, s, t, the sum of their parities can be represented as E + E + E + O + O. The sum of two odd numbers (O + O) is always even (E). The sum of any number of even numbers is always even (E + E + E = E). Consequently, the overall expression simplifies to E + E, which is always even.
Thus, p + q + r – s – t is definitely even. Statement 1 is correct.
Incorrect Options
Statement 2: 2p + q + 2r – 2s + t is definitely odd.
Consider the parity of each term in the expression:
- Any integer multiplied by 2 (an even number) results in an even number. Therefore, 2p, 2r, and 2s are all even (E).
- The expression can be rewritten in terms of parity as: E + q + E – E + t.
The parity of this entire expression is determined solely by the parity of the terms q and t, as the even terms do not change the overall parity (E + q + E – E + t has the same parity as q + t).
We know that among p, q, r, s, t, there are three even and two odd numbers. The variables q and t can assume different parity combinations:
- Case 1: q and t are both even (E, E). In this scenario, q + t = E + E = E (Even).
- Case 2: q and t are both odd (O, O). In this scenario, q + t = O + O = E (Even).
- Case 3: One of q or t is even, and the other is odd (E, O or O, E). In this scenario, q + t = E + O = O (Odd).
Since q + t can result in either an even or an odd number depending on the specific assignment of parities to q and t, the expression 2p + q + 2r – 2s + t is not "definitely odd". For example, if q and t are both odd (e.g., p=2, r=4, s=6, q=1, t=3), then the expression evaluates to 2(2) + 1 + 2(4) – 2(6) + 3 = 4 + 1 + 8 – 12 + 3 = 8, which is an even number.
Thus, Statement 2 is incorrect.