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The positive integer k has exactly two positive prime factor [#permalink]
27 Feb 2008, 19:33

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Difficulty:

95% (hard)

Question Stats:

38% (01:53) correct
63% (01:19) wrong based on 136 sessions

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 d.

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

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

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

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

Answer (C)

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

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

Answer (C)

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. _________________

Re: What is the value of K - Confusing one [#permalink]
23 Sep 2009, 08:24

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This post received KUDOS

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.

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)

Re: What is the value of K - Confusing one [#permalink]
26 Sep 2009, 09:39

3

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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

Re: Positive integer 'K' has exactly two positive prime factors, [#permalink]
07 Nov 2013, 00:02

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

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.

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

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

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)

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....?

Re: What is the value of K - Confusing one [#permalink]
31 Dec 2013, 03:16

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Expert's post

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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.

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)

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.

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

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)

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.

Answer: D.

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?

Re: What is the value of K - Confusing one [#permalink]
02 Jan 2014, 04:19

Expert's post

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.

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)

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.

Answer: D.

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. _________________