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

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Joined: 14 Dec 2017
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Re: The positive integer k has exactly two positive prime factors, 3 and 7  [#permalink]

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11 Jun 2018, 22:34
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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21 Oct 2018, 02:55
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 &nbs [#permalink] 21 Oct 2018, 02:55

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