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16 Dec 2019, 02:39
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C
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Difficulty:
75%
(hard)
Question Stats:
51% (01:51) correct
49%
(01:44)
wrong
based on 217
sessions
History
Date
Time
Result
Not Attempted Yet
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
Are You Up For the Challenge: 700 Level Questions
(1) x is a two-digit number
(2) x^2 has 3 positive factors
Are You Up For the Challenge: 700 Level Questions
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16 Nov 2021, 03:00
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Bunuel
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
General Discussion
Archit3110
Major Poster
Updated on: 16 Dec 2019, 10:04
Originally posted by Archit3110 on 16 Dec 2019, 04:26.
Last edited by Archit3110 on 16 Dec 2019, 10:04, edited 1 time in total.
Last edited by Archit3110 on 16 Dec 2019, 10:04, edited 1 time in total.
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#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
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
Bunuel
