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# The positive integer k has exactly two positive prime

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The positive integer k has exactly two positive prime [#permalink]

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20 Jun 2010, 12:38
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The positive integer k has exactly two positive prime factors 3 and 7. If k has a total of 6 positive factors including 1 and k, what is the value of k?

(1) 9 is a factor of k.
(2) 49 is a factor of k.
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20 Jun 2010, 13:48
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testprep2010 wrote:
The positive integer k has exactly two positive prime factors 3 and 7. If k has a total of 6 positive factors including 1 and k, what is the value of k?

(1) 9 is a factor of k.
(2) 49 is a factor of k.

Finding the Number of Factors of an Integer:

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.

Back to the original question:
"k has exactly two positive prime factors 3 and 7" --> $$k=3^m*7^n$$, where $$m=integer\geq{1}$$ and $$n=integer\geq{1}$$;
"k has a total of 6 positive factors including 1 and k" --> $$(m+1)(n+1)=6$$. Note here that neither $$m$$ nor $$n$$ can be more than 2 as in this case $$(m+1)(n+1)$$ will be more than 6.

So, there are only two values of $$k$$ possible:
1. if $$m=1$$ and $$n=2$$ --> $$k=3^1*7^2=3*49$$;
2. if $$m=2$$ and $$n=1$$ --> $$k=3^2*7^1=9*7$$.

(1) 9 is a factor of k --> we have the second case, hence $$k=3^2*7^1=9*7$$. Sufficient.

(2) 49 is a factor of k --> we have the first case, hence $$k=3^1*7^2=3*49$$. Sufficient.

BUT: in DS statements never contradict, so this cannot be real GMAT question. I guess one of the statements should be "x is NOT factor of k". In this case answer still would be D, but the question will be of GMAT type.

Hope it's clear.
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30 Jun 2010, 05:49
Bunuel wrote:
testprep2010 wrote:
The positive integer k has exactly two positive prime factors 3 and 7. If k has a total of 6 positive factors including 1 and k, what is the value of k?

(1) 9 is a factor of k.
(2) 49 is a factor of k.

Finding the Number of Factors of an Integer:

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.

Back to the original question:
"k has exactly two positive prime factors 3 and 7" --> $$k=3^m*7^n$$, where $$m=integer\geq{1}$$ and $$n=integer\geq{1}$$;
"k has a total of 6 positive factors including 1 and k" --> $$(m+1)(n+1)=6$$. Note here that neither $$m$$ nor $$n$$ can be more than 2 as in this case $$(m+1)(n+1)$$ will be more than 6.

So, there are only two values of $$k$$ possible:
1. if $$m=1$$ and $$n=2$$ --> $$k=3^1*7^2=3*49$$;
2. if $$m=2$$ and $$n=1$$ --> $$k=3^2*7^1=9*7$$.

(1) 9 is a factor of k --> we have the second case, hence $$k=3^2*7^1=9*7$$. Sufficient.

(2) 49 is a factor of k --> we have the first case, hence $$k=3^1*7^2=3*49$$. Sufficient.

BUT: in DS statements never contradict, so this cannot be real GMAT question. I guess one of the statements should be "x is NOT factor of k". In this case answer still would be D, but the question will be of GMAT type.

Hope it's clear.

In the right question

2)49 is NOT a factor of K
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03 Jul 2010, 02:08
Bunuel wrote:

BUT: in DS statements never contradict, so this cannot be real GMAT question. I guess one of the statements should be "x is NOT factor of k". In this case answer still would be D, but the question will be of GMAT type.

Hope it's clear.

Good guess.....took me 10 minutes staring at the question trying to figure out wtf i was missing
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16 Aug 2010, 22:19
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Phew! Glad to see I wasn't missing something ridiculous.

Could one of the mods modify the original question to make the two statements agree with each other a-la Bunuel's correction?
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16 Aug 2010, 22:56
Bunuel wrote:
testprep2010 wrote:
The positive integer k has exactly two positive prime factors 3 and 7. If k has a total of 6 positive factors including 1 and k, what is the value of k?

(1) 9 is a factor of k.
(2) 49 is a factor of k.

Finding the Number of Factors of an Integer:

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.

Back to the original question:
"k has exactly two positive prime factors 3 and 7" --> $$k=3^m*7^n$$, where $$m=integer\geq{1}$$ and $$n=integer\geq{1}$$;
"k has a total of 6 positive factors including 1 and k" --> $$(m+1)(n+1)=6$$. Note here that neither $$m$$ nor $$n$$ can be more than 2 as in this case $$(m+1)(n+1)$$ will be more than 6.

So, there are only two values of $$k$$ possible:
1. if $$m=1$$ and $$n=2$$ --> $$k=3^1*7^2=3*49$$;
2. if $$m=2$$ and $$n=1$$ --> $$k=3^2*7^1=9*7$$.

(1) 9 is a factor of k --> we have the second case, hence $$k=3^2*7^1=9*7$$. Sufficient.

(2) 49 is a factor of k --> we have the first case, hence $$k=3^1*7^2=3*49$$. Sufficient.

BUT: in DS statements never contradict, so this cannot be real GMAT question. I guess one of the statements should be "x is NOT factor of k". In this case answer still would be D, but the question will be of GMAT type.

Hope it's clear.

Bunuel
Back to the contradiction part, so in this case because we are getting 2 different values for k from 1 and 2 - that is why this is not a good gmat question? If (2) said 49 is NOT a factor of k, then looking at (2) alone we could reach the conclusion that k = 63 and 1 already gave us 63, hence D? Is the reasoning correct?
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17 Aug 2010, 04:22
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mainhoon wrote:
Bunuel wrote:
testprep2010 wrote:
The positive integer k has exactly two positive prime factors 3 and 7. If k has a total of 6 positive factors including 1 and k, what is the value of k?

(1) 9 is a factor of k.
(2) 49 is a factor of k.

Finding the Number of Factors of an Integer:

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.

Back to the original question:
"k has exactly two positive prime factors 3 and 7" --> $$k=3^m*7^n$$, where $$m=integer\geq{1}$$ and $$n=integer\geq{1}$$;
"k has a total of 6 positive factors including 1 and k" --> $$(m+1)(n+1)=6$$. Note here that neither $$m$$ nor $$n$$ can be more than 2 as in this case $$(m+1)(n+1)$$ will be more than 6.

So, there are only two values of $$k$$ possible:
1. if $$m=1$$ and $$n=2$$ --> $$k=3^1*7^2=3*49$$;
2. if $$m=2$$ and $$n=1$$ --> $$k=3^2*7^1=9*7$$.

(1) 9 is a factor of k --> we have the second case, hence $$k=3^2*7^1=9*7$$. Sufficient.

(2) 49 is a factor of k --> we have the first case, hence $$k=3^1*7^2=3*49$$. Sufficient.

BUT: in DS statements never contradict, so this cannot be real GMAT question. I guess one of the statements should be "x is NOT factor of k". In this case answer still would be D, but the question will be of GMAT type.

Hope it's clear.

Bunuel
Back to the contradiction part, so in this case because we are getting 2 different values for k from 1 and 2 - that is why this is not a good gmat question? If (2) said 49 is NOT a factor of k, then looking at (2) alone we could reach the conclusion that k = 63 and 1 already gave us 63, hence D? Is the reasoning correct?

Yes, it's not a good GMAT question as the single numerical value of k from (1) differs from the single numerical value of k from (2) - statements contradict each other.

It seems that statement (2) should be: 49 is NOT a factor of k --> we have the second case, hence $$k=3^2*7^1=9*7$$. Sufficient.

So as statement (1) is sufficient and statement (2) is also sufficient, answers is D. In this case each statement gives the same value of k, thus the problem of contradiction is resolved.
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Re: The positive integer k has exactly two positive prime [#permalink]

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20 Oct 2012, 10:36
Thanks Bunuel for clarifying
both statements contradict. kept me going in circle for 4 minutes and then i referred to your explanation.
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Re: The positive integer k has exactly two positive prime [#permalink]

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27 Feb 2013, 04:35
Statement B is (2) 7^2 is not a factor of k.

In that case the example in statement 1 will be the only example that satisfies the condition plus we have to satisfy 6 factors

3^2 * 7^1

(2+1) (1+1) = 6 factors
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Re: The positive integer k has exactly two positive prime [#permalink]

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Re: The positive integer k has exactly two positive prime [#permalink]

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07 Sep 2015, 23:26
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The positive integer k has exactly two positive prime [#permalink]

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22 Dec 2015, 06:14
Bunuel wrote:
testprep2010 wrote:
The positive integer k has exactly two positive prime factors 3 and 7. If k has a total of 6 positive factors including 1 and k, what is the value of k?

(1) 9 is a factor of k.
(2) 49 is a factor of k.

Finding the Number of Factors of an Integer:

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.

Back to the original question:
"k has exactly two positive prime factors 3 and 7" --> $$k=3^m*7^n$$, where $$m=integer\geq{1}$$ and $$n=integer\geq{1}$$;
"k has a total of 6 positive factors including 1 and k" --> $$(m+1)(n+1)=6$$. Note here that neither $$m$$ nor $$n$$ can be more than 2 as in this case $$(m+1)(n+1)$$ will be more than 6.

So, there are only two values of $$k$$ possible:
1. if $$m=1$$ and $$n=2$$ --> $$k=3^1*7^2=3*49$$;
2. if $$m=2$$ and $$n=1$$ --> $$k=3^2*7^1=9*7$$.

(1) 9 is a factor of k --> we have the second case, hence $$k=3^2*7^1=9*7$$. Sufficient.

(2) 49 is a factor of k --> we have the first case, hence $$k=3^1*7^2=3*49$$. Sufficient.

BUT: in DS statements never contradict, so this cannot be real GMAT question. I guess one of the statements should be "x is NOT factor of k". In this case answer still would be D, but the question will be of GMAT type.

Hope it's clear.

Bunuel, why can't either $$m$$ or $$n$$ be $$0$$?
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Re: The positive integer k has exactly two positive prime [#permalink]

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22 Dec 2015, 06:27
Expert's post
TooLong150 wrote:
Bunuel wrote:
testprep2010 wrote:
The positive integer k has exactly two positive prime factors 3 and 7. If k has a total of 6 positive factors including 1 and k, what is the value of k?

(1) 9 is a factor of k.
(2) 49 is a factor of k.

Finding the Number of Factors of an Integer:

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.

Back to the original question:
"k has exactly two positive prime factors 3 and 7" --> $$k=3^m*7^n$$, where $$m=integer\geq{1}$$ and $$n=integer\geq{1}$$;
"k has a total of 6 positive factors including 1 and k" --> $$(m+1)(n+1)=6$$. Note here that neither $$m$$ nor $$n$$ can be more than 2 as in this case $$(m+1)(n+1)$$ will be more than 6.

So, there are only two values of $$k$$ possible:
1. if $$m=1$$ and $$n=2$$ --> $$k=3^1*7^2=3*49$$;
2. if $$m=2$$ and $$n=1$$ --> $$k=3^2*7^1=9*7$$.

(1) 9 is a factor of k --> we have the second case, hence $$k=3^2*7^1=9*7$$. Sufficient.

(2) 49 is a factor of k --> we have the first case, hence $$k=3^1*7^2=3*49$$. Sufficient.

BUT: in DS statements never contradict, so this cannot be real GMAT question. I guess one of the statements should be "x is NOT factor of k". In this case answer still would be D, but the question will be of GMAT type.

Hope it's clear.

Bunuel, why can't either $$m$$ or $$n$$ be $$0$$?

We are told that k has exactly two positive prime factors 3 and 7. Now, if say n is 0, then $$k=3^m*7^0=3^m*1$$ and in this case k would have only one prime 3.

Hope it's clear.
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Re: The positive integer k has exactly two positive prime   [#permalink] 22 Dec 2015, 06:27
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