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Of 128 boxes of oranges, each box contains at least 120 and at most 144 oranges. The number of boxes containing the same number of oranges is at least
- a)5
- b)106
- c)3
- d)9
- e)20
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
Of 128 boxes of oranges, each box contains at least 120 and at most 14...
Each box contains at least 120 and at most 144 oranges.
So boxes may contain 25 different numbers of oranges among 120, 121, 122, .... 144.
Lets start counting.
1st 25 boxes contain different numbers of oranges and this is repeated till 5 sets as 25*5=125.
Now we have accounted for 125 boxes. Still 3 boxes are remaining. These 3 boxes can have any number of oranges from 120 to 144.
Already every number is in 5 boxes. Even if these 3 boxes have different number of oranges, some number of oranges will be in 6 boxes.
Hence the number of boxes containing the same number of oranges is at least 6.
So boxes may contain 25 different numbers of oranges among 120, 121, 122, .... 144.
Lets start counting.
1st 25 boxes contain different numbers of oranges and this is repeated till 5 sets as 25*5=125.
Now we have accounted for 125 boxes. Still 3 boxes are remaining. These 3 boxes can have any number of oranges from 120 to 144.
Already every number is in 5 boxes. Even if these 3 boxes have different number of oranges, some number of oranges will be in 6 boxes.
Hence the number of boxes containing the same number of oranges is at least 6.
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Community Answer
Of 128 boxes of oranges, each box contains at least 120 and at most 14...
To solve this problem, we need to determine the minimum number of boxes that contain the same number of oranges.
Given:
- Each box contains at least 120 and at most 144 oranges.
- There are 128 boxes in total.
Let's assume that the minimum number of boxes containing the same number of oranges is x.
Minimum number of oranges in a box = 120
Maximum number of oranges in a box = 144
We know that the total number of oranges is equal to the sum of oranges in each box. So, we can write the equation:
Total number of oranges = x * minimum number of oranges + (128 - x) * maximum number of oranges
Since the number of oranges in each box must be an integer, the total number of oranges must also be an integer. Therefore, the maximum number of oranges that can be divided by x is the greatest common divisor (GCD) of the minimum and maximum number of oranges.
GCD(120, 144) = 24
So, the total number of oranges must be a multiple of 24.
Now, let's calculate the total number of oranges:
Total number of oranges = 128 * 120 + 0 * 144 (assuming x = 0)
If we increase the value of x by 1, the total number of oranges will increase by 24. So, we need to find the minimum value of x such that the total number of oranges is a multiple of 24.
128 * 120 = 15360
15360 % 24 = 0
The total number of oranges is already a multiple of 24. Therefore, the minimum number of boxes containing the same number of oranges is at least 0.
But the question asks for the minimum number of boxes that contain the same number of oranges, which means we need to find the minimum value of x such that the total number of oranges is a multiple of 24 and x is greater than 0.
To find the minimum value of x, we need to find the factors of 24.
Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
Since x represents the number of boxes containing the same number of oranges, it must be a factor of 24.
Therefore, the minimum value of x is 1.
Hence, the number of boxes containing the same number of oranges is at least 1.
Option C states that the number of boxes containing the same number of oranges is 3, which is greater than 1. Therefore, option C is the correct answer.
Given:
- Each box contains at least 120 and at most 144 oranges.
- There are 128 boxes in total.
Let's assume that the minimum number of boxes containing the same number of oranges is x.
Minimum number of oranges in a box = 120
Maximum number of oranges in a box = 144
We know that the total number of oranges is equal to the sum of oranges in each box. So, we can write the equation:
Total number of oranges = x * minimum number of oranges + (128 - x) * maximum number of oranges
Since the number of oranges in each box must be an integer, the total number of oranges must also be an integer. Therefore, the maximum number of oranges that can be divided by x is the greatest common divisor (GCD) of the minimum and maximum number of oranges.
GCD(120, 144) = 24
So, the total number of oranges must be a multiple of 24.
Now, let's calculate the total number of oranges:
Total number of oranges = 128 * 120 + 0 * 144 (assuming x = 0)
If we increase the value of x by 1, the total number of oranges will increase by 24. So, we need to find the minimum value of x such that the total number of oranges is a multiple of 24.
128 * 120 = 15360
15360 % 24 = 0
The total number of oranges is already a multiple of 24. Therefore, the minimum number of boxes containing the same number of oranges is at least 0.
But the question asks for the minimum number of boxes that contain the same number of oranges, which means we need to find the minimum value of x such that the total number of oranges is a multiple of 24 and x is greater than 0.
To find the minimum value of x, we need to find the factors of 24.
Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
Since x represents the number of boxes containing the same number of oranges, it must be a factor of 24.
Therefore, the minimum value of x is 1.
Hence, the number of boxes containing the same number of oranges is at least 1.
Option C states that the number of boxes containing the same number of oranges is 3, which is greater than 1. Therefore, option C is the correct answer.
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