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22 May 2024, 01:54
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B
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Difficulty:
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Question Stats:
57% (01:06) correct
43%
(01:13)
wrong
based on 23
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If \(8n^2\) has twelve positive factors, how many distinct prime factors does the positive integer \(n\) have?
A. None
B. 1
C. 2
D. 3
E. Cannot be determined from the given information
A. None
B. 1
C. 2
D. 3
E. Cannot be determined from the given information
22 May 2024, 01:54
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Official Solution:
If \(8n^2\) has twelve positive factors, how many distinct prime factors does the positive integer \(n\) have?
A. None
B. 1
C. 2
D. 3
E. Cannot be determined from the given information
For \(8n^2 = 2^3n^2\) to have twelve factors, \(n\) must be a prime other than 2: \(8n^2 = 2^3 * (prime)^2\). In this case, the number of factors can be calculated as \((3+1)(2+1) = 12\).
Another possibility is when \(n = 2^4\): \(8n^2 = 2^3 * (2^4)^2 = 2^{11}\). In this case, the number of factors is also \(11+1 = 12\).
In both cases (\(n = prime\) other than 2 or \(n = 2^4\)), \(n\) has one prime factor.
Answer: B
If \(8n^2\) has twelve positive factors, how many distinct prime factors does the positive integer \(n\) have?
A. None
B. 1
C. 2
D. 3
E. Cannot be determined from the given information
For \(8n^2 = 2^3n^2\) to have twelve factors, \(n\) must be a prime other than 2: \(8n^2 = 2^3 * (prime)^2\). In this case, the number of factors can be calculated as \((3+1)(2+1) = 12\).
Another possibility is when \(n = 2^4\): \(8n^2 = 2^3 * (2^4)^2 = 2^{11}\). In this case, the number of factors is also \(11+1 = 12\).
In both cases (\(n = prime\) other than 2 or \(n = 2^4\)), \(n\) has one prime factor.
Answer: B
02 Sep 2024, 04:43
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\(8n^2 = 2^k * a^x * b^y * ...\)
a, b, ...: prime factors other than 2
\(8n^2\) has 12 positive factors
=> (k+1)(a+1)(b+1)... = 12
k >=3 ----> k + 1 >=4
12 = 12 = 6 * 2 = 4 * 3
k+1 = 12 => n has only 1 prime factor, which is 2
k+1 = 6 => (a+1)(b+1)... = 2 => There can only be 1 more prime factor => n still only has 1 prime factor
k+1 = 4 => (a+1)(b+1)... = 3 => There can only be 1 more prime factor => n still only has 1 prime factor
n can only has 1 prime factor
a, b, ...: prime factors other than 2
\(8n^2\) has 12 positive factors
=> (k+1)(a+1)(b+1)... = 12
k >=3 ----> k + 1 >=4
12 = 12 = 6 * 2 = 4 * 3
k+1 = 12 => n has only 1 prime factor, which is 2
k+1 = 6 => (a+1)(b+1)... = 2 => There can only be 1 more prime factor => n still only has 1 prime factor
k+1 = 4 => (a+1)(b+1)... = 3 => There can only be 1 more prime factor => n still only has 1 prime factor
n can only has 1 prime factor
