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The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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### HideShow timer Statistics 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

Attachment: dsafasfds.JPG [ 14.91 KiB | Viewed 43804 times ]

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.

Originally posted by gmatnub on 27 Feb 2008, 20:33.
Last edited by Bunuel on 04 Jan 2018, 07:54, edited 3 times in total.
Renamed the topic, edited the question and added the OA.
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The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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48
32
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: The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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10
9
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
##### General Discussion
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Posts: 11
Re: The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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1
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: The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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3
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: The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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10
2
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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GMAT 1: 660 Q45 V35 GMAT 2: 710 Q49 V38 Re: The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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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: The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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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....?
Intern  Joined: 04 May 2013
Posts: 44
Re: The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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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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Posts: 56303
Re: The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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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 factors, 3 and 7  [#permalink]

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3
statement (1)
if 3^2 is a factor of k, then so is 3^1.
therefore, we already have four factors: 1, 3^1, 3^2, and 7.
but we also know that (3^1)(7) and (3^2)(7) must be factors, since 3^2 and 7 are both part of the prime factorization of k.
that's already six factors, so we're done: k must be (3^2)(7). if it were any bigger, then there would be more than these six factors.
sufficient.

statement (2)
if 7 is a factor of k, but 7^2 isn't, then the prime factorization of k contains EXACTLY one 7.
therefore, we need to find out how many 3's will produce six factors when paired with exactly one 7.
in fact, it's data sufficiency, so we don't even have to find this number; all we have to do is realize that adding more 3's will always increase the number of factors, so, there must be exactly one number of 3's that will produce the correct number of factors. (as already noted above, that's two 3's, or 3^2.)
sufficient.
Hence D.
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Re: The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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1
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?

K > 0
K = $$3^a$$*$$7^b$$
(a+1)(b+1)=6
K = ?

Quote:
(1) 3^2 is a factor of k
(2) 7^2 is NOT a factor of k

1) $$3^2$$ is a factor of K
=> (2+1)(b+1)=6
=> b = 1
We can find the value of K.
Sufficient.

2) $$7^2$$ is NOT a factor of K
(a+1)(b+1)=6
=> b≠ 2 ; b can only be 1.
=> a = 2, b = 1
We can find the value of K.
Sufficient.

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Re: The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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Bunuel,

Are there other good questions to practice this concept you could point me to?

Thanks a lot,

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Re: The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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Re: The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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Fantastic thank you.

I also found your incredible thread on tips and hints for specific quant topics! Looks like I have a full day worth of material ahead of me. Thanks!
https://gmatclub.com/forum/tips-and-hin ... l#p1379270

Bunuel wrote:

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Re: The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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1
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

We are given that K has two positive prime factors, 3 and 7, and that K has a total of 6 factors including 1 and K. Thus, we know that the factors of K include 1, 3, 7, 21, and K. We must determine the value of K.

Statement One Alone:

3^2 is a factor of K.

Let’s list the factors of K: 1, 3, 7, 21, K, and 3^2 = 9

Since 3 and 7 are factors of K, 3 x 7 = 21 must also be a factor of K. Similarly, since 9 and 7 are both factors of K (and they are relatively prime), 9 x 7 = 63 must also be a factor of K. Since we already have 6 factors, K must equal 63. Statement one alone is sufficient to answer the question.

Statement Two Alone:

7^2 is not a factor of K.

If 7 is a factor of K but 7^2 is not a factor of K, then 3^2 = 9 must be a factor of K (otherwise, K has only 4 factors, namely 1, 3, 7, and 21). If 9 is a factor of K, then the list of factors of K is 1, 3, 7, 9, 21, 63. Therefore, K = 63. Statement two alone is also sufficient to answer the question.

Alternative solution:

We need to determine the value of K. We are given that K has two positive prime factors, 3 and 7. Therefore, the prime factorization of K must be K = 3^m x 7^n for some positive integers m and n greater than 1. Recall that the total number of factors of a number can be obtained by multiplying the numbers resulting from adding 1 to the exponents in the prime factorization. Thus, the total number of factors of K is (m + 1) x (n + 1). Since we are given that K has a total of 6 factors, (m + 1) x (n + 1) = 6. Since m and n are both greater than 1, either m = 2 and n = 1 OR m = 1 and n = 2.

Statement One Alone:

3^2 is a factor of K.

This tells us that m = 2, so n = 1. Therefore, K = 3^2 x 7^1 = 63.

Statement one alone is sufficient to answer the question.

Statement Two Alone:

7^2 is not a factor of K.

This tells us that n ≠ 2, so n = 1, and thus m = 2. Therefore, K = 3^2 x 7^1 = 63.

Statement two alone is sufficient to answer the question.

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Re: The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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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

Given $$K = 3^x * 7^y$$ & $$(x+1)(y+1) = 6$$

Hence x can take 2 values to give integer values of y
$$x = 2$$, we get $$y = 1$$ & hence $$K = 3^2 * 7$$
$$x = 1$$, we get $$y = 2$$ & hence $$K = 3 * 7^2$$

Statement 1:
$$3^2$$ is a factor of $$K$$
Hence $$y = 1$$ & $$K = 3^2 * 7$$

Statement 1 alone is Sufficient.

Statement 2:
$$7^2$$ is NOT a factor of $$K$$
Hence $$x = 2$$ & $$y = 1$$
$$K = 3^2 * 7$$

Statement 2 alone is Sufficient.

Thanks,
GyM
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Re: The positive integer k has exactly two positive prime factors  [#permalink]

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sonusaini1 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) \small 3^{2} is a factor of K.
(2) \small 7^{2} is not a factor of K.

A Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D EACH statement ALONE is sufficient.
E Statements (1) and (2) TOGETHER are NOT sufficient.

Since there are very few options (k has only 6 positive factors and 2 prime factors), we can just write them all out.
This is an Alternative approach.

First, we know k has 1,3,7,21 and k as distinct factors, meaning that we're missing exactly one. (k can't be 21 because then 1,3,7,21 would be all the positive factors...)

(1) this must be our last positive factor and is therefore sufficient. (In particular, it means that k is the LCM of 1,3,7,9, and 21, which is 9*7 = 63)
Sufficient.

(2) this is equivalent to statement (1): if we can't increase the power of 7, we must increase the power of 3 (because these are the only prime factors), so 3^2 is a factor of k.
Sufficient

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Re: The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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Given: K=3^m*7^n & (m+1)*(n+1)=6
So we can derive that either m+1 or n+1 is 2 or 3

Statement 1: 3^2 is a factor of k. Therefore m+1=2 and n+1=3
Sufficient

Statement 2: 7^2 isn't a factor of k. Therefore m+1=2 and n+1=3
Sufficient

D

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Re: The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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Hi Bunuel ,

I understand the formula how to get the number of factors. But how can you identify each one?

For example, I did prime factorization on 300 = 2^2 * 5^2 * 3^1 --> The formula for factors therefore is --> (3)(3)(2) = 18.
How can I enumerate the 18 factors then? I know there's 1, 300, 2, 5, 3, but how do I get the others in between? Is there a formula or I just have to do a manual enumeration?

I hope you can help out! Thanks. Re: The positive integer k has exactly two positive prime factors, 3 and 7   [#permalink] 18 Jul 2019, 18:08

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# The positive integer k has exactly two positive prime factors, 3 and 7  