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

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

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New post 09 Mar 2016, 12:40
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A
B
C
D
E

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62% (01:54) correct 38% (01:36) wrong based on 185 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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New post 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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New post 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
Answer B
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Re: Positive integer x has n factors; 3x has 3 factors; Which of the follo  [#permalink]

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New post 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.
Correct answer is B.
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Re: Positive integer x has n factors; 3x has 3 factors; Which of the follo  [#permalink]

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

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New post 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.

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