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A crate contains 103 defective cellphones. If you distribute defective cellphones to \(x\) technicians such that each technician gets the same number of defective cellphones, the number of cellphones left in the crate is 2 less than the number of technicians. Which of the following cannot be the value of \(x\)?
A. 3
B. 5
C. 7
D. 14
E. 15
A. 3
B. 5
C. 7
D. 14
E. 15
02 Sep 2024, 09:04
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Official Solution:
A crate contains 103 defective cellphones. If you distribute defective cellphones to \(x\) technicians such that each technician gets the same number of defective cellphones, the number of cellphones left in the crate is 2 less than the number of technicians. Which of the following cannot be the value of \(x\)?
A. 3
B. 5
C. 7
D. 14
E. 15
Assuming each technician gets \(n\) cellphones, we get \(xn + (x - 2) = 103\). This simplifies to \(x(n + 1) = 105\), which implies that \(x\), the number of technicians, must be a factor of 105, such that \(x - 2 \geq 0\) (after all, there cannot be a negative number of cellphones left in the crate after the distribution of the cellphones). From the options, only option D, 14, is not a factor of 105, making D the correct answer.
Answer: D
A crate contains 103 defective cellphones. If you distribute defective cellphones to \(x\) technicians such that each technician gets the same number of defective cellphones, the number of cellphones left in the crate is 2 less than the number of technicians. Which of the following cannot be the value of \(x\)?
A. 3
B. 5
C. 7
D. 14
E. 15
Assuming each technician gets \(n\) cellphones, we get \(xn + (x - 2) = 103\). This simplifies to \(x(n + 1) = 105\), which implies that \(x\), the number of technicians, must be a factor of 105, such that \(x - 2 \geq 0\) (after all, there cannot be a negative number of cellphones left in the crate after the distribution of the cellphones). From the options, only option D, 14, is not a factor of 105, making D the correct answer.
Answer: D
