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05 Oct 2019, 15:03
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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:
38% (03:01) correct
62%
(02:35)
wrong
based on 21
sessions
History
Date
Time
Result
Not Attempted Yet
A certain number 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
(A) 31
(B) 45
(C) 49
(D) 63
(E) 87
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05 Oct 2019, 19:19
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Hovkial
A certain number 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
(A) 31
(B) 45
(C) 49
(D) 63
(E) 87
Formula: Number of factors of N = 2^a*3^b*5^c*7^d*. . . = (a+1)(b+1)(c+1)(d+1)......
Note: The formula is independent of any prime number as BASE. It only depends on the power(s) of prime number(s).
Possible solutions for 16 factors are
Case1: If N= 2^1*3^1*5^1*7^1 = (1+1)(1+1)(1+1)(1+1) = 16
Square of N = 2^2*3^2*5^2*7^2
Number of factors = 3*3*3*3 = 81
Case2: If N= 2^1*3^1*5^3 = (1+1)(1+1)(3+1) = 16
Square of N = 2^2*3^2*5^6
Number of factors = 3*3*7 = 63
Case3: If N= 2^1*3^7 = (1+1)(7+1) = 16
Square of N = 2^2*3^14
Number of factors = 3*15 = 45
Case4: If N= 2^3*3^3 = (3+1)(3+1) = 16
Square of N = 2^6*3^6
Number of factors = 7*7 = 49
Case4: If N= 2^15 = (15+1) = 16
Square of N = 2^30
Number of factors = 30+1 = 31
IMO Option E
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Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Problem Solving (PS) Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
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