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# A positive integer N has exactly 6 unique positive factors. What is th

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A positive integer N has exactly 6 unique positive factors. What is th  [#permalink]

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21 Mar 2019, 22:22
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85% (hard)

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33% (01:44) correct 68% (01:34) wrong based on 40 sessions

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A positive integer N has exactly 6 unique positive factors. What is the highest prime factor of N?

(1) 6 is a factor of N.
(2) 9 is a factor of N.
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Re: A positive integer N has exactly 6 unique positive factors. What is th  [#permalink]

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29 Mar 2019, 08:28
amanvermagmat wrote:
A positive integer N has exactly 6 unique positive factors. What is the highest prime factor of N?

(1) 6 is a factor of N.
(2) 9 is a factor of N.

Question involving 'number of factors = x' are almost always solved via basic number properties.
We'll look for such a solution, a Logical approach.

(1) if 6 is a positive factor of N then so are 1,2, and 3 so we have at least 4 factors (1,2,3,6). If a different prime number p were a factor, then we would also have 2p,3p and 6p, too many. So all factors of N must be powers of 2 or 3 and the highest prime factor of N is 3.
Sufficient.

(2) then (1,3,9) are factors of N. If we also have p other than 3, then we would have (1,3,9,p,3p,9p) which is fine. So we can have p = 2 like before or p > 3. Then we cannot answer the question.
Insufficient.

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Re: A positive integer N has exactly 6 unique positive factors. What is th  [#permalink]

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29 Mar 2019, 18:04
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amanvermagmat wrote:
A positive integer N has exactly 6 unique positive factors. What is the highest prime factor of N?

(1) 6 is a factor of N.
(2) 9 is a factor of N.

Since N has exactly 6 unique positive factors. It can be expressed as either $$z^5$$ or $$x^1*y^2$$ because number of factor is (p+1)(q+1).... where $$x=y^p*z^q*....$$

Statement 1:
6 is a factor of N. So, both 2 and 3 are factors of N. Which means N can be expressed as either $$2^1*3^2$$ or$$2^2*3^1$$. In both case highest prime factor is 3. Sufficient.

Statement 2:

9 is a factor of N. Now N can be either written as $$3^5$$ or $$3^2*x^1$$, where x is the other prime factor. x can be greater than 3 or not. So, we can't say for sure which is greatest prime factor. Insufficient.

IMO, Option A.
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Re: A positive integer N has exactly 6 unique positive factors. What is th   [#permalink] 29 Mar 2019, 18:04
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