If x is a positive integer, what is the number of different positive

Question

If x is a positive integer, what is the number of different positive factors of 24x ?

(1) x is a two-digit number
(2) x^2 has 3 positive factors

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Archit3110 · Answer

#1
x is a two-digit number
24* x
many possible values 
insufficient
#2
x^2 has 3 positive factors

only possible when x is prime ; 
so factor of 24 * x ; 
possible to determine different + factors of 24*x
2^3*3^1 * 2 ; 5*2 ; 10 or 2^3*3^2 *3 ; 4*4 ; 16
insufficient
from 1 &2
x= 11,13,. any prime no 
total factors of 24*x ; 2^3*3^1*x^1 ; 8+2 ; 10
IMO C

nitish13 · Answer

IMO C

Statement 1 alone insufficient

Statement 2 shows X prime 
X can be 2 or 3 
Hence Statement 2 not sufficient.

However, if Statement 1 + 2
X is prime different from 2 or 3.
Hence can determine number of positive factors.

dsan6422 · Answer

If x is a positive integer, what is the number of different positive factors of 24x ?
(1) x is a two-digit number
(2) x^2 has 3 positive factors

Question: 24x =2^3 . 3 . x

statement 1: insufficient 
statement 2: x^2 has 3 positive factors
 We have to realize that here the questions is talking about  factors and Not Prime Factors 
 This means that  two of these 3 factors  is  1 and the x^2 itself.  
 ---> x must be a prime number because the square of prime is the only integer that has only 3 factors
 
 24x = 2^3.3.P where P is any prime number
 This is not sufficient because the answer is going to be different depending on wether P= 2 or 3 or some other P numbers

Statement 1 + Statement 2
 Sufficient because now we now that the Prime number is different from 2 and 3, and now we can use the formulas for finding the number of
 of factors of 2^3 . 3 . P = (3+1)(2)(2)

Answer C

Bunuel · Answer

Official Solution:

If  x  is a positive integer, what is the number of different positive factors of  24x  ? 
 
Factorize:  24x=2^3*3*x 
 
(1)  x  is a two-digit number
 
Clearly insufficient.
 
(2)  x^2  has 3 positive factors
 
The above means that  x  is a prime number. Only the squares of primes ( p^2 ) have three factors: 1,  p , and  p^2 
 
If  x=11  ( or any other prime except 2 and 3 ), then  24x=2^3*3*11  will have  (3+1)(1+1)(1+1)=16  factors; 
 
If  x=2 , then  24x=2^4*3  will have  (4+1)(1+1)=10  factors. 
 
If  x=3 , then  24x=2^3*3^2  will have  (3+1)(2+1)=12  factors. 
 
Not sufficient.
 
(1)+(2) Since (1) says that  x  is a two-digit number, then  x  cannot be 2 or 3, so it must be a two-digit prime. For any two digit prime,  24x=2^3*3*x  will have  (3+1)(1+1)(1+1)=16  factors. Sufficient.

Answer: C

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If x is a positive integer, what is the number of different positive

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