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If \(x\) is a positive integer, what is the number of different positive factors of \(39x\) ?
(1) \(x\) is a two-digit number
(2) \(x^2\) has 3 positive factors
(1) \(x\) is a two-digit number
(2) \(x^2\) has 3 positive factors
09 Nov 2021, 07:42
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Official Solution:
If \(x\) is a positive integer, what is the number of different positive factors of \(39x\) ?
Factorize: \(39x=3*13*x\)
(1) \(x\) is a two-digit number
Clearly insufficient.
(2) \(x^2\) has 3 positive factors
The above means that \(x\) is a prime number. Only the squares of primes (\(p^2\)) have three factors: 1, \(p\), and \(p^2\)
If \(x=11\) (or any other prime except 3 and 13), then \(39x=3*11*13\) will have \((1+1)(1+1)(1+1)=8\) factors;
If \(x=13\), then \(39x=3*13^2\) will have \((1+1)(2+1)=6\) factors.
Not sufficient.
(1)+(2) Examples we used for (2) (\(x=11\) and \(x=13\)) also satisfy (1), so even taken together the statements are not sufficient to get the answer: \(39x\) can have 8 or 6 factors. Not sufficient.
Answer: E
If \(x\) is a positive integer, what is the number of different positive factors of \(39x\) ?
Factorize: \(39x=3*13*x\)
(1) \(x\) is a two-digit number
Clearly insufficient.
(2) \(x^2\) has 3 positive factors
The above means that \(x\) is a prime number. Only the squares of primes (\(p^2\)) have three factors: 1, \(p\), and \(p^2\)
If \(x=11\) (or any other prime except 3 and 13), then \(39x=3*11*13\) will have \((1+1)(1+1)(1+1)=8\) factors;
If \(x=13\), then \(39x=3*13^2\) will have \((1+1)(2+1)=6\) factors.
Not sufficient.
(1)+(2) Examples we used for (2) (\(x=11\) and \(x=13\)) also satisfy (1), so even taken together the statements are not sufficient to get the answer: \(39x\) can have 8 or 6 factors. Not sufficient.
Answer: E
