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21 Mar 2024, 02:20
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List K consist of n consecutive positive integers: \(\{1, 2, 3, ..., n\}\). If the list contains exactly 38 multiples of 13, what is the maximum number of multiples of 11 it can contain ?
A. 40
B. 44
C. 45
D. 46
E. 47
A. 40
B. 44
C. 45
D. 46
E. 47
21 Mar 2024, 02:20
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Official Solution:
List K consist of n consecutive positive integers: \(\{1, 2, 3, ..., n\}\). If the list contains exactly 38 multiples of 13, what is the maximum number of multiples of 11 it can contain ?
A. 40
B. 44
C. 45
D. 46
E. 47
We aim to maximize the number of multiples of 11 in the list while ensuring there are exactly 38 multiples of 13. To achieve this, we can design a list where \(n\) is the closest multiple of 11 that is less than the 39th multiple of 13. The 39th multiple of 13 is \(39*13 = 507\). The closest multiple of 11 less than 507 is 506 (to find this, observe that 440 and 66 are both multiples of 11, hence \(440 + 66 = 506\), is also a multiple of 11.). Therefore, \(n= 506\). In this scenario, the list will contain exactly 38 multiples of 13 and the maximum number of multiples of 11 possible.
In this setup, the number of multiples of 11 in the list would be:
\(\frac{last \ multiple \ of \ 11 \ in \ the \ list - first \ multiple \ of \ 11 \ in \ the \ list}{11} +1 = \frac{506 - 11}{11} +1 = 46\).
Answer: D
List K consist of n consecutive positive integers: \(\{1, 2, 3, ..., n\}\). If the list contains exactly 38 multiples of 13, what is the maximum number of multiples of 11 it can contain ?
A. 40
B. 44
C. 45
D. 46
E. 47
We aim to maximize the number of multiples of 11 in the list while ensuring there are exactly 38 multiples of 13. To achieve this, we can design a list where \(n\) is the closest multiple of 11 that is less than the 39th multiple of 13. The 39th multiple of 13 is \(39*13 = 507\). The closest multiple of 11 less than 507 is 506 (to find this, observe that 440 and 66 are both multiples of 11, hence \(440 + 66 = 506\), is also a multiple of 11.). Therefore, \(n= 506\). In this scenario, the list will contain exactly 38 multiples of 13 and the maximum number of multiples of 11 possible.
In this setup, the number of multiples of 11 in the list would be:
\(\frac{last \ multiple \ of \ 11 \ in \ the \ list - first \ multiple \ of \ 11 \ in \ the \ list}{11} +1 = \frac{506 - 11}{11} +1 = 46\).
Answer: D
24 Mar 2024, 11:17
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i feel like the question statement is a little ambigious, cause a person can think of 'N' to be 38 x 13
