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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Jeffery Miller

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