Let be the statement ' is a prime number' and be the statement ' has exactly two positive divisors'. Which of the following is correct? (Here is a positive integer.)
Answer and explanation
Correct answer: A. is both a necessary and a sufficient condition for
- Step 1: A prime has exactly the divisors and itself, so .
- Step 2: A number with exactly two positive divisors has divisors and only (with ), so it is prime: .
Answer
is both a necessary and a sufficient condition for .
Why not the others: has only one divisor and so satisfies neither statement; no integer satisfies exactly one of them.
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