The positive integer k has exactly two positive prime factor : GMAT Data Sufficiency (DS)
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# The positive integer k has exactly two positive prime factor

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

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27 Feb 2008, 19:33
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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) 3^2 is a factor of k
(2) 7^2 is NOT a factor of k

[Reveal] Spoiler:
I searched thru 6-7 pages using keywords, but I did not find this question asked, I think this could be a newly added question in the gmatprep software.

somewhat of a tricky wording question, especially when time is running short. oa is d.

correction: oa is D.
[Reveal] Spoiler: OA

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Last edited by Bunuel on 03 Nov 2013, 05:08, edited 2 times in total.
Renamed the topic, edited the question and added the OA.
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Re: Gmatprep DS: the positive integer k has exactly two [#permalink]

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27 Feb 2008, 20:34
Based on stem, the 6 factors of k are 1,3,7,21, x and k . where 7 < x < k.

If statement (1) is used, the factors are, 1, 3, 7, 9, 21, k. k = 63. sufficient
Since stem says 3,7 are the only prime factors, x has to be 3^2 since x cannot be 7^2. - sufficient

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Re: Gmatprep DS: the positive integer k has exactly two [#permalink]

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28 Feb 2008, 06:19
gmatnub wrote:
Gmatprep DS: 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) 3^2 is a factor of k

2) 7^2 is NOT a factor of k

I searched thru 6-7 pages using keywords, but I did not find this question asked, I think this could be a newly added question in the gmatprep software.

somewhat of a tricky wording question, especially when time is running short. oa is a.

K has 6 factors: 1,3,7,21,X,K (different factors) Essentially we need to find X then we will know K.

1: X must be 9. b/c K has two 3's as factors.

2: if 7^2 is not a factor of K then X cannot be 49. Since we only have 3 and 7 as prime factors, 3 must be the other factor and X would be 9.

I get D

Im not sure why OA is A... =(
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Re: Gmatprep DS: the positive integer k has exactly two [#permalink]

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16 Feb 2009, 21:55
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Good question. I have a different way of solving this.

Let P1 = Power of first factor
Let P2 = Power of second factor
The number of factors can be found using the equation (P1 + 1)(P2 + 1). This is a rule, I didn't come up with this.
Therefore here we have:
2*3 or 3*2, both equal 6.

statement 1: says that 3*2 is out, therefore sufficient
statement 2: says that 3*2 is out, therefore sufficient.

note that we cannot use 6*1, because then we have a 7^0 or a 3^0, which is not the case here.

What do you think?
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Re: Gmatprep DS: the positive integer k has exactly two [#permalink]

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19 Feb 2009, 10:07
x1050us wrote:
Based on stem, the 6 factors of k are 1,3,7,21, x and k . where 7 < x < k.

If statement (1) is used, the factors are, 1, 3, 7, 9, 21, k. k = 63. sufficient
Since stem says 3,7 are the only prime factors, x has to be 3^2 since x cannot be 7^2. - sufficient

I don't understand why k=63, why can't it be 27 (due to 3 x 9)??
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Re: Gmatprep DS: the positive integer k has exactly two [#permalink]

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19 Feb 2009, 11:11
DaveGG wrote:
x1050us wrote:
Based on stem, the 6 factors of k are 1,3,7,21, x and k . where 7 < x < k.

If statement (1) is used, the factors are, 1, 3, 7, 9, 21, k. k = 63. sufficient
Since stem says 3,7 are the only prime factors, x has to be 3^2 since x cannot be 7^2. - sufficient

I don't understand why k=63, why can't it be 27 (due to 3 x 9)??

In that case, k would have 3^3 as factor. If so, the k would have more than 6 factors as under: 1, 3, 7, 9, 21, 27, 42, 63, and 189

gmatnub wrote:
Gmatprep DS: 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) 3^2 is a factor of k
2) 7^2 is NOT a factor of k

We need one more either 3 or 7 to have 6 +ve factors of k.

a: 3^2 makes 6 +ve factors.
b. if there is no 7^2 as a factor of k, then it also makes sure that 3^3 is a factor of k.
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Re: What is the value of K - Confusing one [#permalink]

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23 Sep 2009, 08:24
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From the stem, we know that K's factors are 1, 3, 7, 21 (3*7), __, and K.

1) This tells us there are two factors of 3, so 9 is also a factor of K. K's factors are 1, 3, 7, 9, 21, and K. Since there are two 3's and a 7 in K's factors, then 3*3*7 = 63 is also a factor.

Therefore K's factors are 1, 3, 7, 9, 21, 63.
SUFFICIENT

2) If there are not 2 7's in K's factors, and there are exactly 6 factors total, there must be two factors of 3. Otherwise, if we were to use a non-prime factor, then K would have more than 6 factors. (Remember 'K' has exactly two positive prime factors)

Therefore, K's factors are 1, 3, 7, 9, 21, 63.
SUFFICIENT

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Re: What is the value of K - Confusing one [#permalink]

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26 Sep 2009, 09:39
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Positive integer 'K' has exactly two positive prime factors, 3 and 7. If 'K' has a total of 6 factors, including 1 and 'K', what is the value of 'K'?

(1) 3^2 is a factor of 'K'

(2) 7^2 is not a factor of 'K'.

Soln:
Since k has two positive prime factors
k = 3^a * 7^b
k has a total of 6 factors meaning
(a+1) * (b+1) = 6
this can be either
(a+1) * (b+1) = 1 * 6
or
(a+1) * (b+1) = 2 * 3

1 * 6 is not possible because one of the factors will become 0. In tat case k will have just one prime factor. Hence the only option is 2 * 3
So when a = 2, b = 1 and when a = 1, b = 2
thus k can be either 3^2 * 7^1 or 3^1 * 7^2

Now considering statement 1 alone,
3^2 is a factor of 'K'. This will be true only when k = 3^2 * 7^1
Thus statement 1 alone is sufficient

Now considering statement 2 alone,
7^2 is not a factor of 'K'. This will be true only when k = 3^2 * 7^1
Thus statement 2 alone is sufficient

Hence D
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Re: Positive integer 'K' has exactly two positive prime factors, [#permalink]

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07 Nov 2013, 00:02
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Re: The positive integer k has exactly two positive prime factor [#permalink]

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09 Nov 2013, 16:06
gmatnub 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) 3^2 is a factor of k
(2) 7^2 is NOT a factor of k

[Reveal] Spoiler:
I searched thru 6-7 pages using keywords, but I did not find this question asked, I think this could be a newly added question in the gmatprep software.

somewhat of a tricky wording question, especially when time is running short. oa is d.

correction: oa is D.

K=3^a * 7^b

and (a+1) *(b+1) = 6

so either a=1, b=2, or a=2, b=1

both statements would help us get K=3^2 * 7= 63
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Re: The positive integer k has exactly two positive prime factor [#permalink]

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16 Nov 2013, 13:43
gmatnub 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) 3^2 is a factor of k
(2) 7^2 is NOT a factor of k

The solutions that try to name each factor are dangerous because one can always run the risk to overlook one or two factors. Oddly enough, I feel that the best way to approach this problem is through "combinatories"! It is just a matter of seeing that the total number of factors in K (6 as mentioned in the stem) is the product of the "group of possible factors including 3" and "the group of possible factors including 7".

Statement one is sufficient: As per the statement, the group of possible factors including 3 is 3 (0, 1 or 2 times) - therefore 3 possibilities. We do know that total number of factors of K is 6, so the group of possible factors including 7 has to be two - when 7 appears 0 or 1 time. So group of three - three elements (0,1 or 2) times group of 7 - two elements (0 or 1) equals 6!

Statement two is also sufficient: The only possible factors of K is 6, so either "the group of factors including 7" is two (7^1) or three (7^2) possibilities. The statement rules out the later, that leaves you with two possibilities for "the group of factors including 7".
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Re: What is the value of K - Confusing one [#permalink]

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30 Dec 2013, 18:14
samiam7 wrote:
From the stem, we know that K's factors are 1, 3, 7, 21 (3*7), __, and K.

1) This tells us there are two factors of 3, so 9 is also a factor of K. K's factors are 1, 3, 7, 9, 21, and K. Since there are two 3's and a 7 in K's factors, then 3*3*7 = 63 is also a factor.

Therefore K's factors are 1, 3, 7, 9, 21, 63.
SUFFICIENT

2) If there are not 2 7's in K's factors, and there are exactly 6 factors total, there must be two factors of 3. Otherwise, if we were to use a non-prime factor, then K would have more than 6 factors. (Remember 'K' has exactly two positive prime factors)

Therefore, K's factors are 1, 3, 7, 9, 21, 63.
SUFFICIENT

The bold part is what I do not understand. I am sorry, but I dont get the factors part where it says "there are two 3's and a 7 in K's factors".

Can someone please explain why is this the case? What allows us to say this? I mean what allows us to say two 3's and a 7? 9 is 3^2, 21 is 3*7, but....?
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Re: What is the value of K - Confusing one [#permalink]

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31 Dec 2013, 03:16
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jjack0310 wrote:
samiam7 wrote:
From the stem, we know that K's factors are 1, 3, 7, 21 (3*7), __, and K.

1) This tells us there are two factors of 3, so 9 is also a factor of K. K's factors are 1, 3, 7, 9, 21, and K. Since there are two 3's and a 7 in K's factors, then 3*3*7 = 63 is also a factor.

Therefore K's factors are 1, 3, 7, 9, 21, 63.
SUFFICIENT

2) If there are not 2 7's in K's factors, and there are exactly 6 factors total, there must be two factors of 3. Otherwise, if we were to use a non-prime factor, then K would have more than 6 factors. (Remember 'K' has exactly two positive prime factors)

Therefore, K's factors are 1, 3, 7, 9, 21, 63.
SUFFICIENT

The bold part is what I do not understand. I am sorry, but I dont get the factors part where it says "there are two 3's and a 7 in K's factors".

Can someone please explain why is this the case? What allows us to say this? I mean what allows us to say two 3's and a 7? 9 is 3^2, 21 is 3*7, but....?

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:

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?

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

(2) 7^2 is NOT a factor of k --> we have the second case, hence $$k=3^2*7^1=9*7$$. Sufficient.

Hope it's clear.
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Re: What is the value of K - Confusing one [#permalink]

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01 Jan 2014, 08:58
Bunuel wrote:
jjack0310 wrote:
From the stem, we know that K's factors are 1, 3, 7, 21 (3*7), __, and K.

1) This tells us there are two factors of 3, so 9 is also a factor of K. K's factors are 1, 3, 7, 9, 21, and K. Since there are two 3's and a 7 in K's factors, then 3*3*7 = 63 is also a factor.

Therefore K's factors are 1, 3, 7, 9, 21, 63.
SUFFICIENT

2) If there are not 2 7's in K's factors, and there are exactly 6 factors total, there must be two factors of 3. Otherwise, if we were to use a non-prime factor, then K would have more than 6 factors. (Remember 'K' has exactly two positive prime factors)

Therefore, K's factors are 1, 3, 7, 9, 21, 63.
SUFFICIENT

The bold part is what I do not understand. I am sorry, but I dont get the factors part where it says "there are two 3's and a 7 in K's factors".

Can someone please explain why is this the case? What allows us to say this? I mean what allows us to say two 3's and a 7? 9 is 3^2, 21 is 3*7, but....?

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:

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?

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

(2) 7^2 is NOT a factor of k --> we have the second case, hence $$k=3^2*7^1=9*7$$. Sufficient.

Hope it's clear.

Thank you much Bunuel.

Just one last question, and the reason that we are not acounting for the case when m = 0, and n = 5 is because 3^0 or 7^0 would be 1, and in that case, 3 is not a prime factor of k. Correct?
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Re: What is the value of K - Confusing one [#permalink]

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02 Jan 2014, 04:19
jjack0310 wrote:
Bunuel wrote:
jjack0310 wrote:
From the stem, we know that K's factors are 1, 3, 7, 21 (3*7), __, and K.

1) This tells us there are two factors of 3, so 9 is also a factor of K. K's factors are 1, 3, 7, 9, 21, and K. Since there are two 3's and a 7 in K's factors, then 3*3*7 = 63 is also a factor.

Therefore K's factors are 1, 3, 7, 9, 21, 63.
SUFFICIENT

2) If there are not 2 7's in K's factors, and there are exactly 6 factors total, there must be two factors of 3. Otherwise, if we were to use a non-prime factor, then K would have more than 6 factors. (Remember 'K' has exactly two positive prime factors)

Therefore, K's factors are 1, 3, 7, 9, 21, 63.
SUFFICIENT

The bold part is what I do not understand. I am sorry, but I dont get the factors part where it says "there are two 3's and a 7 in K's factors".

Can someone please explain why is this the case? What allows us to say this? I mean what allows us to say two 3's and a 7? 9 is 3^2, 21 is 3*7, but....?

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:

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?

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

(2) 7^2 is NOT a factor of k --> we have the second case, hence $$k=3^2*7^1=9*7$$. Sufficient.

Hope it's clear.

Thank you much Bunuel.

Just one last question, and the reason that we are not acounting for the case when m = 0, and n = 5 is because 3^0 or 7^0 would be 1, and in that case, 3 is not a prime factor of k. Correct?

Absolutely, m and n must be greater than zero because if they are not then 3 and 7 are not the factors of k.
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Re: The positive integer k has exactly two positive prime factor [#permalink]

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23 Feb 2015, 13:44
Hello from the GMAT Club BumpBot!

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

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24 Feb 2015, 06:06
Stem says that

3^x*7^y=k

(x+1)*(y+1)=6 with at least one 3 and 7
xy+x+y+1=6
x(y+1)+y=5
only possibility is x=2 and y=1, so k=3^2*7=63

No need in any statement

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

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29 Feb 2016, 09:26
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Re: The positive integer k has exactly two positive prime factor [#permalink]

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08 Jan 2017, 21:39
Given : $$k=3^n * 7^m$$ --------(1), where m & n are powers of prime factors, 3 and 7.
also we know that k has a total of 6 positive factors, including 1 and k

this can be represented as (n+1)(m+1)=6 --------(2)

statement (1) : 3^2 is a factor of k
n=2 , substitute in (2) , we get m=1
put n=2 & m=1 in (1) , we get $$k=3^2 * 7^1$$ >> k=63 >> sufficient.

statement (2) : $$7^2$$ is NOT a factor of k
as from the question stem , we know that 7 is among the prime factors of k, hence ,the minimum power of 7 is 1.
therefore m=1 , substitute in (2) , we get n=2
put n=2 & m=1 in (1) , we get $$k=3^2 * 7^1$$ >> k=63 >> sufficient.

Ans : D
Re: The positive integer k has exactly two positive prime factor   [#permalink] 08 Jan 2017, 21:39
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