It is given that if a positive integer is a perfect square then is divisible by 4 or is odd. It is also given that is not divisible by 4. Which of the following must be true?

Answer and explanation

Correct answer: A. If is a perfect square, then is odd

  1. Step 1: Let : ' is a perfect square', : 'divisible by 4', : 'odd'. We know and .
  2. Step 2: If holds then holds; since is false, must hold. So (but itself need not be true).
Answer
If is a perfect square, then is odd.
Why not the others: ' is odd' and ' is not a perfect square' both go beyond what is known, because may be an even non-square such as or a non-square odd number; 'if is a perfect square then is not odd' and ' is a perfect square and not odd' contradict the conclusion .

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