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# Positive integer x has n factors; 3x has 3 factors; Which of the follo

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Joined: 02 Sep 2009
Posts: 47037
Positive integer x has n factors; 3x has 3 factors; Which of the follo [#permalink]

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09 Mar 2016, 12:40
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55% (hard)

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61% (01:15) correct 39% (01:11) wrong based on 180 sessions

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Positive integer x has n factors; 3x has 3 factors; Which of the following values can n take?

I. 1
II. 2
III. 3

A. I only
B. II only
C. I or II
D. II or III
E. I or III

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Re: Positive integer x has n factors; 3x has 3 factors; Which of the follo [#permalink]

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09 Mar 2016, 21:06
I would choose 'B'

Given statement is that 3x has three factors which means 3x is a perfect square of 3. hence x=3

3 has 2 factors, making n=2
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Re: Positive integer x has n factors; 3x has 3 factors; Which of the follo [#permalink]

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09 Mar 2016, 22:18
1
Squares have odd number of factors.The square which is a multiple of 3 is 9.
3 has factors 1,3 .
Therefore 2 factors
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Re: Positive integer x has n factors; 3x has 3 factors; Which of the follo [#permalink]

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09 Mar 2016, 23:19
2
Bunuel wrote:
Positive integer x has n factors; 3x has 3 factors; Which of the following values can n take?

I. 1
II. 2
III. 3

A. I only
B. II only
C. I or II
D. II or III
E. I or III

Given: (i) $$x$$ is an integer; (ii) $$x > 0$$; and (iii) $$3x$$ has 3 factors
Rule: Only perfect squares have odd number of factors. Only perfect squares of prime numbers have exactly 3 factors.
Therefore, $$x$$ must be equal to 3 for $$3x$$ to be a perfect square of a prime number. So, $$x$$ has exactly 2 factors.
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Re: Positive integer x has n factors; 3x has 3 factors; Which of the follo [#permalink]

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14 Mar 2016, 05:33
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Posts: 2669
Re: Positive integer x has n factors; 3x has 3 factors; Which of the follo [#permalink]

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26 Sep 2017, 16:37
Bunuel wrote:
Positive integer x has n factors; 3x has 3 factors; Which of the following values can n take?

I. 1
II. 2
III. 3

A. I only
B. II only
C. I or II
D. II or III
E. I or III

Notice that 3x must be a perfect square since only perfect squares have an odd number of factors. Furthermore, 3x must be the square of a prime number since only squares of prime numbers have 3 factors. That is, if p is a prime, then p^2 has 3 factors, namely, 1, p, and p^2.

Since 3x is the square of a prime number, we see that x must be 3 so that 3x = 9. Since x = 3 and 3 has 2 factors, then n must be 2.

Alternate Solution:

Let’s go over Roman numerals I and III:

Roman Numeral I: x has only 1 factor

The only positive integer with 1 factor is 1 itself; but 3(1) = 3 does not have 3 factors. This is impossible.

Roman Numeral III: x has 3 factors

If x is a number with 3 factors, then 3x will have more than 3 factors. For instance, x = 9 has 3 factors and 3x = 27 has 4 factors. x = 4 has 3 factors and 3x = 12 has 6 factors. This is impossible as well.

The only remaining possibility is n = 2.

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Re: Positive integer x has n factors; 3x has 3 factors; Which of the follo   [#permalink] 26 Sep 2017, 16:37
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