Let be the statement ' is a multiple of 4' and be the statement ' is a multiple of 8'. Which of the following is correct? (Here is a positive integer.)

Answer and explanation

Correct answer: D. is both a necessary and a sufficient condition for

  1. Step 1: If then , a multiple of : .
  2. Step 2: Conversely, if is not a multiple of , then with odd or is odd. Then with odd (or is odd), which is not a multiple of . So .
Answer
is both a necessary and a sufficient condition for .
Why not the others: A tempting counterexample would be a number like or , but their squares ( and ) are not multiples of , so they do not break .

Practise more Necessary and sufficient conditions.

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