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Positive integer k

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by sanju09 » Fri Feb 07, 2014 1:30 am
Winner2013 wrote:I have a doubt about this question and need some help. I guess this doubt is really stupid but still I would appreciate if someone can help.

The question clearly says that the number k has exactly 2 positive prime factors - 3 and 7. now for 63- after factorization we get

63= 3*3*7

so we have 3 prime factors here right? - 3,3,7. Then how does the question say only 2 prime factors and the answer comes out to be 63?

am i interpreting the question in a wrong way? if a prime factor(3) is repeated as in case of 63, what do we say about how many prime factors does the number have?

please help.

thanks,
Pooja
While taking the distinct prime factors of any positive integer greater than 1 into account, we never consider the index number to count, for example 2^4.5^6 has exactly two prime factors, 2 and 5.
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by Anaira Mitch » Sun Jan 29, 2017 4:24 pm
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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by Scott@TargetTestPrep » Thu Feb 02, 2017 11:18 am
success1111 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.

Answer: D

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