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16 Nov 2021, 04:24
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If \(p\) and \(q\) are integers and \(\frac{pq}{10}\) is an integer, is \(\frac{p}{5}\) an integer?
(1) \(\frac{q}{2}\) is an integer
(2) \(p\) is a prime number
(1) \(\frac{q}{2}\) is an integer
(2) \(p\) is a prime number
16 Nov 2021, 04:24
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Official Solution:
If \(p\) and \(q\) are integers and \(\frac{pq}{10}\) is an integer, is \(\frac{p}{5}\) an integer?
(1) \(\frac{q}{2}\) is an integer.
The above means that \(q\) is even.
Can \(p\) be a multiple of 5? Certainly. Consider \(p=5\) and \(q=2=even\);
Can \(p\) NOT be a multiple of 5? Certainly. Consider \(p=1\) and \(q=10=even\).
Not sufficient.
(2) \(p\) is a prime number
Can \(p\) be a multiple of 5? Certainly. Consider \(p=5=prime\) and \(q=2\);
Can \(p\) NOT be a multiple of 5? Certainly. Consider \(p=3=prime\) and \(q=10\).
Not sufficient.
(1)+(2) Examples used for (2) are also valid for (1) (because we consider even values of \(q\) there). So, even taken together we cannot answer the question. Not sufficient.
Answer: E
If \(p\) and \(q\) are integers and \(\frac{pq}{10}\) is an integer, is \(\frac{p}{5}\) an integer?
(1) \(\frac{q}{2}\) is an integer.
The above means that \(q\) is even.
Can \(p\) be a multiple of 5? Certainly. Consider \(p=5\) and \(q=2=even\);
Can \(p\) NOT be a multiple of 5? Certainly. Consider \(p=1\) and \(q=10=even\).
Not sufficient.
(2) \(p\) is a prime number
Can \(p\) be a multiple of 5? Certainly. Consider \(p=5=prime\) and \(q=2\);
Can \(p\) NOT be a multiple of 5? Certainly. Consider \(p=3=prime\) and \(q=10\).
Not sufficient.
(1)+(2) Examples used for (2) are also valid for (1) (because we consider even values of \(q\) there). So, even taken together we cannot answer the question. Not sufficient.
Answer: E
