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Can the positive integer k be expressed as the product of [#permalink]
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11 Feb 2012, 19:08
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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
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Re: Integer K [#permalink]
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11 Feb 2012, 20:39
Statement 1: K^2 has one more factor. Let's say if K was a 4 (nonprime ) then it will have 4 factors. [1,2,4]. If you square 4, 16 has 5[1,2,4,8,16] factors. The factors increased by 2. The only way for a factor to increase by 1 is if the number is prime. 13 has 2 factors [1,13]. 169 has 3 factors [1,13,169].
This mean K is a prime. The only 2 numbers that can multiple to 13 is 13 and 1. Therefore we know the answer for the question is NO.
Answer A is sufficient



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Re: Integer K [#permalink]
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11 Feb 2012, 21:24
I think the question lacks a bit of information. It mentions that both the integers should be greater than 1 but does not mention that both integers should be different. However, still, when prime numbers are squared, the number of different factors increases by 1 which is a product of the prime no multiplied by the prime number.
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Re: Integer K [#permalink]
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12 Feb 2012, 01:04
Can the positive integer k be expressed as the product of two integers, each of which is greater than 1?Question basically asks whether k is a prime number. If it is, then it cannot be expressed as the product of two integers, each of which is greater than 1 (definition of a prime number). (1) k^2 has one more positive factor than k > if k is a prime then it has 2 factors: 1, and k > k^2 will have one more, so 3 factors: 1, k, and k^2. If k is some composite number greater than 1, then it has more than 2 factors and # of factors of k^2 will increase by more than just by 1 (try any composite number to check this). If k=1 then k^2 will have the same # of factor as k: one. Hence k=prime. Sufficient. (2) 11 < k < 19. k can be 13, so prime, as well as 14 so not a prime. Not sufficient. Answer: A. Hope it's clear.
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Re: Can the positive integer k be expressed as the product of [#permalink]
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13 Feb 2012, 15:57
Can the positive integer k be expressed as the product of two integers, each of which is greater than 1? Question basically asks whether k is a prime number. If it is, then it cannot be expressed as the product of two integers, each of which is greater than 1 (definition of a prime number). Bunuel  I take it you mean product of two DIFFERENT integers, each of which is greater than 1.
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Re: Can the positive integer k be expressed as the product of [#permalink]
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13 Feb 2012, 16:02
If k is some composite number greater than 1, then it has more than 2 factors > So K will be prime or not? Am I reading something wrong Bunuel?
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Re: Can the positive integer k be expressed as the product of [#permalink]
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13 Feb 2012, 17:51
enigma123 wrote: If k is some composite number greater than 1, then it has more than 2 factors > So K will be prime or not? Am I reading something wrong Bunuel? A prime number is a natural number with exactly two distinct natural number divisors: 1 and itself. Otherwise a number is called a composite number.So, composite numbers are not primes. We are considering 3 cases for (1): k=prime; k=composite>1; K=1; And get that k can be only a prime number. enigma123 wrote: Can the positive integer k be expressed as the product of two integers, each of which is greater than 1?
Question basically asks whether k is a prime number. If it is, then it cannot be expressed as the product of two integers, each of which is greater than 1 (definition of a prime number).
Bunuel  I take it you mean product of two DIFFERENT integers, each of which is greater than 1. It doesn't really matter. Can you express a prime as the product of two same integers?
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Can the positive integer [m]k[/m] be expressed [#permalink]
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28 Oct 2012, 07:47
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.



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Re: Can the positive integer [m]k[/m] be expressed [#permalink]
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28 Oct 2012, 18:59
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danzig 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. Note, for a number K the factors are K,1 and 'few other' depending upon whether it is prime or not for number k^2  the factors are 1, k and k^2 and 'few other' The 'few other' factors depend on the fact whether K is divisible by a number or not. For example: factors of 5 : 1, 5 and ?  nothing else factors of 6: 1, 6 and ?  (2,3) Similarly, factors of 25: 1,5, 25 and ?  nothing else ( because 5 is not divisible by anything else and hence can not be broken into any other number) factors of 36: 1,6,36 and ?  (2,3,4,9, 12,18) Therefore if you notice the pattern only for a prime number, number of factors of k^2 is one more than number of factors for k. Now with this concept target the question stem 1 : it shows us k is prime using above mentioned concept. can a prime number be a product of 2 integers each greather than 1? no. So we have a sufficient statement to say No. stem 2: K could be anything from 12 to 18. So ans would be if k is 12 then yes, but if k is 13 then no. hence insufficient. Therefore ans A (only statement 1 is sufficient) it is. Hope it is clear.
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Re: Can the positive integer [m]k[/m] be expressed [#permalink]
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28 Oct 2012, 22:28
danzig 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. Basically, what the question asks is whether k is not a prime number. 1)k is prime. Sufficient. 2)k can be 12,13,14,15,16,17,18. Both primes and non primes appear in this set. Insufficient Answer is A. Kudos Please... If my post helped.
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Re: Can the positive integer [m]k[/m] be expressed [#permalink]
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Re: Can the positive integer k be expressed as the product of [#permalink]
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01 Aug 2017, 23:32
kys123 wrote: Statement 1: K^2 has one more factor. Let's say if K was a 4 (nonprime ) then it will have 4 factors. [1,2,4]. If you square 4, 16 has 5[1,2,4,8,16] factors. The factors increased by 2. The only way for a factor to increase by 1 is if the number is prime. 13 has 2 factors [1,13]. 169 has 3 factors [1,13,169].
This mean K is a prime. The only 2 numbers that can multiple to 13 is 13 and 1. Therefore we know the answer for the question is NO.
Answer A is sufficient K may or may not be prime. Who said anything about the factor other than k being not equal to 1?
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Re: Can the positive integer k be expressed as the product of
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