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

Question

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

Divyadisha · Answer

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

vardhanindaram · Answer

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

nalinnair · Answer

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 .

stonecold · Answer

Excellent Question..
her the only value of x possible is 3
so x=3 
hence the number of factors of 3= 1,3 => two 
hence B is correct

JeffTargetTestPrep · Answer

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

abhimanyuarya · Answer

x has n factors

3x has 3 factors

3 factors of a number are only possible in case of (prime)^2

Therefore, 3x = 9 = 3^2 i.e. 3 factors

X = 3 and number of factors of 3 are 2 numbers.

This is the only possibility among answer choices.

Akkiiii · Answer

What i did was 
I took x as 2 
this way I got the answer B but I just wanted to know if the way of solving it was right
x=2, so this way 3*x has 3 factors
as 3*2 also has 3 factors= 1,3,2

Just wanted to know whether solving it this way is right or wrong

Thanks a bunch

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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

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