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21 May 2024, 01:30
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E
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Difficulty:
65%
(hard)
Question Stats:
59% (01:41) correct
41%
(02:01)
wrong
based on 170
sessions
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If the greatest common factor of the integers \(\frac{n}{11}\) and \(\frac{n}{13}\) is 6, what is the number of positive factors of the positive integer \(n\)?
A. 2
B. 4
C. 6
D. 8
E. 16
A. 2
B. 4
C. 6
D. 8
E. 16
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02 Jun 2024, 17:30
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Official Solution:
If the greatest common factor of the integers \(\frac{n}{11}\) and \(\frac{n}{13}\) is 6, what is the number of positive factors of the positive integer \(n\)?
A. 2
B. 4
C. 6
D. 8
E. 16
\(n\) must be a multiple of 2 and 3. Because \(6=2*3\) is a factor of \(\frac{n}{11}\) and \(\frac{n}{13}\), then \(6=2*3\) must also be a factor of \(n\). Additionally, notice that \(n\) cannot have higher powers of 2 or 3 because in this case the GCF would contain higher powers of these primes;
\(n\) must be a multiple of 11 because we are told that \(\frac{n}{11}\) is an integer. \(n\) cannot have a higher power of 11 because in this case the GCF would contain 11;
\(n\) must be a multiple of 13 because we are told that \(\frac{n}{13}\) is an integer. \(n\) cannot have a higher power of 13 because in this case the GCF would contain 13;
\(n\) cannot be a multiple of any other prime because in this case that prime would also appear in \(\frac{n}{11}\) and \(\frac{n}{13}\) and thus in the GCF.
Thus, \(n\) must be \(2*3*11*13\). Therefore, \(n\) has \((1+1)(1+1)(1+1)(1+1)=16\) positive factors.
Answer: E
If the greatest common factor of the integers \(\frac{n}{11}\) and \(\frac{n}{13}\) is 6, what is the number of positive factors of the positive integer \(n\)?
A. 2
B. 4
C. 6
D. 8
E. 16
\(n\) must be a multiple of 2 and 3. Because \(6=2*3\) is a factor of \(\frac{n}{11}\) and \(\frac{n}{13}\), then \(6=2*3\) must also be a factor of \(n\). Additionally, notice that \(n\) cannot have higher powers of 2 or 3 because in this case the GCF would contain higher powers of these primes;
\(n\) must be a multiple of 11 because we are told that \(\frac{n}{11}\) is an integer. \(n\) cannot have a higher power of 11 because in this case the GCF would contain 11;
\(n\) must be a multiple of 13 because we are told that \(\frac{n}{13}\) is an integer. \(n\) cannot have a higher power of 13 because in this case the GCF would contain 13;
\(n\) cannot be a multiple of any other prime because in this case that prime would also appear in \(\frac{n}{11}\) and \(\frac{n}{13}\) and thus in the GCF.
Thus, \(n\) must be \(2*3*11*13\). Therefore, \(n\) has \((1+1)(1+1)(1+1)(1+1)=16\) positive factors.
Answer: E
General Discussion
21 May 2024, 02:06
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Bunuel
\(n = 11 * 13 * 6\)
\(n = 11^1 * 13^1 * 2^1 * 3 ^1\)
Number of factors \(= 2 * 2 * 2 * 2 = 16\)
Option E
