Can the positive integer k be expressed as the product of two integers, each of which is greater than 1?
(1) k^2 has one more positive factor than k.
(2) 11 < k < 19
I don't understand well this explanation of the OE. Please, your help:
The only types of numbers k such that k2 has exactly one more positive factor than k are primes. Prime numbers have two factors and their squares have three. If k had more than two factors, the number of factors would increase by more than 1 when squared. Thus, k must be prime, answering the question.
Product of two integers
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.metallicafan wrote:Can the positive integer k be expressed as the product of two integers, each of which is greater than 1?
(1) k^2 has one more positive factor than k.
(2) 11 < k < 19
I don't understand well this explanation of the OE. Please, your help:
The only types of numbers k such that k2 has exactly one more positive factor than k are primes. Prime numbers have two factors and their squares have three. If k had more than two factors, the number of factors would increase by more than 1 when squared. Thus, k must be prime, answering the question.
Here we have to find if k is a prime number.
(1) k² has one more positive factor than k.
If k is a prime then it has 2 factors, 1 and k implies k² has 3 factors: 1, k, and k².
If k = 1 (not a prime number), then k² will have the same number of factor as k.
Hence k has to be a prime; SUFFICIENT.
(2) 11 < k < 19.
If k = 12, it is not a prime.
If k = 13, it is prime; NOT Sufficient.
The correct answer is A.
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- anuprajan5
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Anurag,
Isn't the question asking us to prove that k is non-prime?
Can the positive integer k be expressed as the product of two integers, each of which is greater than 1
I get the reasoning in statement 1 and 2. But the rephrase of the question seems to be wrong.
Isn't the question asking us to prove that k is non-prime?
Can the positive integer k be expressed as the product of two integers, each of which is greater than 1
I get the reasoning in statement 1 and 2. But the rephrase of the question seems to be wrong.
Regards
Anup
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Both rephrased target questions ("Is k prime?" and "Is k non-prime?") are essentially the same in that a statement that is sufficient to answer one rephrased target question will be sufficient to answer the other rephrased target question.anuprajan5 wrote:Anurag,
Isn't the question asking us to prove that k is non-prime?
Can the positive integer k be expressed as the product of two integers, each of which is greater than 1
I get the reasoning in statement 1 and 2. But the rephrase of the question seems to be wrong.
For example, if we are able to conclude, with certainty, that k is prime, then we can also conclude, with certainty, that k is not non-prime.
Cheers,
Brent