For integers \(m\) and \(n\), both greater than 1, consider the following three statements : \(P : m\) divides \(n\) \(Q : m\) divides \(n^2\) \(R : m\) is prime, then
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- \(Q \wedge R \rightarrow P\)
- \(P \wedge Q \rightarrow R\)
- \(Q \rightarrow R\)
- \(Q \rightarrow P\)
The Correct Option is A
Solution and Explanation
(1) Suppose \(Q\) and \(R\) hold: \(m\) is prime and \(m \mid n^2 = n\cdot n\). A prime dividing a product must divide a factor, so \(m \mid n\), i.e. \(P\) holds. So \(Q \wedge R \rightarrow P\) is always true.
(2) Take \(m=n=4\): \(P\) and \(Q\) are true but \(R\) is false. So \(P \wedge Q \rightarrow R\) fails.
(3) Take \(m=4,\ n=2\): \(4 \mid 4\), so \(Q\) is true but \(R\) is false. \(Q \rightarrow R\) fails.
(4) With the same \(m=4,\ n=2\): \(Q\) is true but \(P\) (\(4 \mid 2\)) is false. \(Q \rightarrow P\) fails.
Answer: (1) \(Q \wedge R \rightarrow P\)
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