A tennis ball factory packages its newly produced balls in either

Question

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

M18-05

Bunuel · Answer

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

hloonker · Answer

hi is there any way in which i can solve this problem without brut force. i still got the right answer '1' - but is there a more fool proof method?

Adarsh_24 · Answer

Alternative method.
Multiples of 25 under 94 => 25, 50, 75
Multiples of 17 under 94=> 17,34,51,68,85
We need 94. Multiples of 25 ends with 5 or 0. So to get 94 we need multiple of 17 to end with 9 or 4.
Only available 34 => 34 plus none of 25, 50, 75 give 94.

Let’s try 93=> 17 multiple should end with 8 or 3.
68
68+25=93
So minimum unboxed 1.

Vivek1707 · Answer

Since 94 is not divisible by either 17 or 25, another way to find the closest number which would be divisible by both would be to keep subtracting 17 from 94 and look for the closest number to 25 or a multple of 25 that results from this subtraction.

94 - 17 = 77
77 - 17 = 60 
60 - 17 = 43 
43 - 17 = 26
26 - 17 = 9

Notice that 26 is the closest multiple of 25
Hence the mininum number of boxes left would be 26 - 25 i.e 1 Answer B

A tennis ball factory packages its newly produced balls in either

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