Question: Is p^{2}+q^{2}+q odd, where p, q are positive integers?<br>Statement I: 2p+q is odd.<br>Statement II: q-2p is odd.

For the next ten (10) items that follow:<br>Each item contains a Question followed by two Statements. Answer each item using the following instructions:

  1. A. Choose this option if the Question can be answered by one of the Statements alone but not by the other.
  2. B. Choose this option if the Question can be answered by either Statement alone.
  3. C. Choose this option if the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone.
  4. D. Choose this option if the Question cannot be answered even by using both Statements together. ✓

Correct Answer: D. Choose this option if the Question cannot be answered even by using both Statements together.

Explanation

Rewrite p^2+q^2+q as p^2 + q(q+1). The term q(q+1) is always even. Thus, the expression is odd if and only if p is odd. Statements I and II only confirm that q is odd (since 2p is even). They provide no information about the parity of p. Cannot be answered.

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