Re: A tennis ball factory packages its newly produced balls in either
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29 Apr 2023, 09:11
Official Solution:
A tennis ball factory packages its newly produced balls in either large boxes, which hold 25 balls each, or small boxes, containing 17 balls each. If there are 94 freshly made tennis balls to be stored, what is the minimum number of balls that will remain unboxed?
A. 0
B. 1
C. 2
D. 3
E. 4
Is it possible to place all 94 balls with no balls remaining unboxed?
• 94 is neither a multiple of 25 nor a multiple of 17.
• If we use one 25-ball box, we'll be left with 94 - 1(25) = 69 balls. 69 is not a multiple of 17. Hence, this case is not possible.
• If we use two 25-ball boxes, we'll be left with 94 - 2(25) = 44 balls. 44 is not a multiple of 17. Hence, this case is not possible either.
• If we use three 25-ball boxes, we'll be left with 94 - 3(25) = 19 balls. 19 is not a multiple of 17. Hence, this case is not possible either.
Since none of the possible combinations of 17-ball and 25-ball boxes can store all 94 balls without any remaining unboxed, it is not possible to place all 94 balls with no balls left unboxed.
Is it possible to place all 93 balls with no balls remaining unboxed?
• 93 is neither a multiple of 25 nor a multiple of 17.
• If we use one 25-ball box, we'll be left with 93 - 1(25) = 68 balls. 68 is a multiple of 17 (68 = 17 * 4). Hence, this case is possible.
• In this case, we would need one 25-ball box and four 17-ball boxes to store all 93 balls without any remaining unboxed.
Therefore, the minimum number of balls that will remain unboxed when storing 94 freshly made tennis balls is 1.
Essentially, we need to find the non-negative integers x and y such that the sum \(25x + 17y\) is closest to, but not greater than, 94. It turns out that when \(x = 1\) and \(y = 4\), the sum equals 93, which is 1 less than 94. Therefore, the minimum number of balls that will remain unboxed is 1.
Answer: B