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29 Dec 2021, 09:45
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
35% (02:21) correct
65%
(02:15)
wrong
based on 93
sessions
History
Date
Time
Result
Not Attempted Yet
A certain positive integer has a total of 16 factors. The square of this number cannot have which one of the following total factors?
(A) 31
(B) 45
(C) 49
(D) 63
(E) 87
Are You Up For the Challenge: 700 Level Questions
(A) 31
(B) 45
(C) 49
(D) 63
(E) 87
Are You Up For the Challenge: 700 Level Questions
29 Dec 2021, 12:42
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If a number is represented in the form \(p^x * q^y * r^z ..\) where p,q,r and x, y , z are positive integers
Total number of factors = (x+1)(y+1)(z+1)
Is given that x consists of total 16 factor.
16 = \(2^4\), thus the total number of factors can be formed by multiplying the 2's in any order.
Lets assume that the number is N
For ex-
16 could have been formed by multiplying 4 * 4, or 2 * 8 or may be just 16* 1.
Option analysis
A) 31
N = \(p^{15}\)
Total factors of N will be 16.
\(N ^ 2 \) = \(p^{(2*15)}\)
Total factors of \(N^2\) will be 31.
Hence this is possible
B) 45
N = \(p^1 * q^7\)
Total factors N will be 2 * 8 = 16.
\(N ^ 2 \) =\(p^2 * q^{14}\)
Total factors of \(N^{2}\) will be 3 * 15 = 45.
Hence this is possible
C) 49
N = \(p^3 * q^3\)
Total factors N will be 4 * 4 = 16.
\(N ^ 2 \) =\(p^6 * q^{6}\)
Total factors of \(N^{2}\) will be 7 *7 = 49.
Hence this is possible
D) 63
N = \(p^1 * q^1 * r^3\)
Total factors N will be 2 * 2 * 4 = 16.
\(N ^ 2 \) =\(p^2 * q^2 * r^6\)
Total factors of \(N^{2}\) will be 3 * 3 * 7= 63.
Hence this is possible
D) 87
Only remaining option
IMO - E
Total number of factors = (x+1)(y+1)(z+1)
Is given that x consists of total 16 factor.
16 = \(2^4\), thus the total number of factors can be formed by multiplying the 2's in any order.
Lets assume that the number is N
For ex-
16 could have been formed by multiplying 4 * 4, or 2 * 8 or may be just 16* 1.
Option analysis
A) 31
N = \(p^{15}\)
Total factors of N will be 16.
\(N ^ 2 \) = \(p^{(2*15)}\)
Total factors of \(N^2\) will be 31.
Hence this is possible
B) 45
N = \(p^1 * q^7\)
Total factors N will be 2 * 8 = 16.
\(N ^ 2 \) =\(p^2 * q^{14}\)
Total factors of \(N^{2}\) will be 3 * 15 = 45.
Hence this is possible
C) 49
N = \(p^3 * q^3\)
Total factors N will be 4 * 4 = 16.
\(N ^ 2 \) =\(p^6 * q^{6}\)
Total factors of \(N^{2}\) will be 7 *7 = 49.
Hence this is possible
D) 63
N = \(p^1 * q^1 * r^3\)
Total factors N will be 2 * 2 * 4 = 16.
\(N ^ 2 \) =\(p^2 * q^2 * r^6\)
Total factors of \(N^{2}\) will be 3 * 3 * 7= 63.
Hence this is possible
D) 87
Only remaining option
IMO - E
19 Dec 2024, 04:57
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Hello from the GMAT Club BumpBot!
Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
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Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
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