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# Can the positive integer k be expressed as the product of

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Can the positive integer k be expressed as the product of [#permalink]

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11 Feb 2012, 20:08
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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
[Reveal] Spoiler: OA

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11 Feb 2012, 21:39
Statement 1: K^2 has one more factor. Let's say if K was a 4 (non-prime ) 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.

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11 Feb 2012, 22:24
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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12 Feb 2012, 02:04
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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.

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, 16:57
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, 17:02
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, 18:51
Expert's post
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, 08:47
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, 19:59
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, 23:28
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

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Re: Can the positive integer [m]k[/m] be expressed [#permalink]

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29 Oct 2012, 04:29
Expert's post
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.

Merging similar topics. Please refer to the solutions above.
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Re: Can the positive integer k be expressed as the product of [#permalink]

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Re: Can the positive integer k be expressed as the product of   [#permalink] 07 Oct 2015, 17:59
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