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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
59% (02:07) correct
41%
(01:52)
wrong
based on 264
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A positive integer N has 27 factors. Which of the following can't be the number of factors of N^2?
A. 53
B. 85
C. 125
D. 144
E. None
A. 53
B. 85
C. 125
D. 144
E. None
Bunuel
To attain the number of factors, we take the power of each prime factor, add one to each, and then take the product of the result.
Ex:
number of factors of 100:
Prime factorization of 100 = 2^2 • 5^2
(2 + 1)(2 + 1) = (3)(3) = 9 factors
We are told that N has 27 factors. This means that the prime factorization of N could be:
x^26 —> (26 + 1) = 27
x^2 • y^8 —> (2 + 1)(8 + 1) = (3)(9) = 27
x^2 • y^2 • z^2 -> (2+1)(2+1)(2+1)=(3)(3)(3)= 27
Number of factors of N^2 could be:
(x^26)^2 = x^52 = (52 + 1) = 53
(x^2 • y^8)^2 = x^4 • y^16 = (4+1)(16 +1) =
(5)(17) = 85
(x^2 • y^2 •z^2)^2 = x^4 • y^4 • z^4 = (4+1)(4+1)(4+1) = (5)(5)(5) = 125
We can eliminate (E) because 53, 85, and 125 can be the number of factors of N^2.
This leaves us with D. 144 cannot be the number of factors of N^2.
IMO Option D
Posted from my mobile device
14 Jan 2025, 18:37
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N squared is not prime. Only prime numbers has even factors. So N squared cannot have even factor 144. Ans (D)
