The positive integer k has exactly two positive prime

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.

Answer: D.

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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In the right question

2)49 is NOT a factor of K

Good guess.....took me 10 minutes staring at the question trying to figure out wtf i was missing

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?

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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Thanks Bunuel for clarifying
both statements contradict. kept me going in circle for 4 minutes and then i referred to your explanation.
+1 Kudos.....

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

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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May i ask you one question?
As i understand that " k has a total of 6 positive factors which includes 1 and k, + exactly two positive prime factors, 3 and 7 " which mean factors k can be (1,3,3,3,7,k) or (1,3,3,7,7,k) or (1,3,7,7,7,k)
So, (1): 9 is a factor of k --> we can have two cases: (1,3,3,3,7,63) or (1,3,3,7,7,441) -> insufficient
same for (2): -> insufficient
Combine: Sufficient:
Please correct me if i misunderstand the point here
Thank alot
P/s: Oh i just realized that there is another factor here is 21 (3*7), I'm sorry, you are correct !
Thanks

Option B is Incorrect. It should be

(B) 49 is NOT a factor of k.

Same question is available mentioning NOT in B

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