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

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Kudos [?]: 1051 [0], given: 5

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

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New post 11 Dec 2017, 17:36
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.

Answer: D
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Kudos [?]: 1051 [0], given: 5

Re: The positive integer k has exactly two positive prime factors, 3 and 7   [#permalink] 11 Dec 2017, 17:36
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The positive integer k has exactly two positive prime factors, 3 and 7

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