A kindergarten has m identical blue balls and 126 identical red balls. One way to store all the blue balls is to distribute them equally among 18 identical cardboard boxes. There are 6 other ways in which all the blue balls can be distributed equally among identical cardboard boxes that are greater than 1; the number of identical cardboard boxes used is different in each of these ways. If the kindergarten manager also wants to distribute the red balls equally among the identical cardboard boxes previously used only to store the blue balls such that no red ball is left out, what is the minimum total number of the balls that a cardboard box will store?a)7b)10c)18d)30e)Cannot be DeterminedCorrect answer is option 'B'. Can you explain this answer?

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A kindergarten has m identical blue balls and 126 identical red balls. One way to store all the blue balls is to distribute them equally among 18 identical cardboard boxes. There are 6 other ways in which all the blue balls can be distributed equally among identical cardboard boxes that are greater than 1; the number of identical cardboard boxes used is different in each of these ways. If the kindergarten manager also wants to distribute the red balls equally among the identical cardboard boxes previously used only to store the blue balls such that no red ball is left out, what is the minimum total number of the balls that a cardboard box will store?

a)
7
b)
10
c)
18
d)
30
e)
Cannot be Determined

Correct answer is option 'B'. Can you explain this answer?

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A kindergarten has m identical blue balls and 126 identical red balls....

Given:

The number of red balls, say r = 126
The number of blue balls = m
m is completely divisible by 18
- m is also divisible by 6 other numbers greater than 1
- So, total factors of m = {1, 18, 6 other factors greater than 1}
- So, total number of factors of m = 2 + 6 = 8

To Find: When 126 and m balls are distributed equally among cardboard boxes, what is the minimum number of (red + blue) balls per box?

Approach:

The minimum number of balls will happen when the number of cardboard boxes used is the maximum
So, to answer the question, we first need to find the maximum number of cardboard boxes that can be used

The 126 red balls can be distributed equally among a number of boxes only if the number of boxes is a factor of 126
- For example, can you split 126 balls equally between 2 boxes? Yes.
- Because 2 is a factor of 126
- Can you split 126 balls equally between 10 boxes? No. Because 10 is not a factor of 126.
Similarly, m blue balls can be distributed equally among a number of boxes only if the number of boxes is a factor of m
Since both red and blue balls need to be distributed equally, the number of boxes must be a common factor of m and 126
So, maximum number of boxes = GCD(m, 126)

2. To find GCD(m, 26), we need to know the value of m.

We’ll use the clues given about m:
- m is of the form 18k
- The total number of factors of m is 8

Working out:

Finding the value of m
- We are given that m is of the form 18k
  - 18 = 2*3²
  - So, m = 2*3²*k
  - So, m has a minimum of 2 prime factors

Also, m has 8 factors
- Either 8 = (0+1)(7+1)
  - That is, m is of the form P
  - Rejected, because we’ve noted above that m definitely has at least 2 prime factors
Or 8 = (1+1)(3+1)
- That is, m is of the form P₁*P₂³
- We already know that m definitely does contain 2 and 3
- So, the only possible value of m is 2*3³

Thus, m = 2*3

Finding GCD(m, 126)
- 126 = 2*3²*7
- As inferred above, m = 2*3³
- So, GCD(m, 126) = 2*3² = 18
Getting to the final answer
- The maximum number of cardboard boxes that can be used = 18
  In this case, the number of:

So, the total number of balls per box = 7 +3 = 10
Looking at the answer choices, we see that the correct answer is Option B

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A kindergarten has m identical blue balls and 126 identical red balls. One way to store all the blue balls is to distribute them equally among 18 identical cardboard boxes. There are 6 other ways in which all the blue balls can be distributed equally among identical cardboard boxes that are greater than 1; the number of identical cardboard boxes used is different in each of these ways. If the kindergarten manager also wants to distribute the red balls equally among the identical cardboard boxes previously used only to store the blue balls such that no red ball is left out, what is the minimum total number of the balls that a cardboard box will store?a)7b)10c)18d)30e)Cannot be DeterminedCorrect answer is option 'B'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about A kindergarten has m identical blue balls and 126 identical red balls. One way to store all the blue balls is to distribute them equally among 18 identical cardboard boxes. There are 6 other ways in which all the blue balls can be distributed equally among identical cardboard boxes that are greater than 1; the number of identical cardboard boxes used is different in each of these ways. If the kindergarten manager also wants to distribute the red balls equally among the identical cardboard boxes previously used only to store the blue balls such that no red ball is left out, what is the minimum total number of the balls that a cardboard box will store?a)7b)10c)18d)30e)Cannot be DeterminedCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A kindergarten has m identical blue balls and 126 identical red balls. One way to store all the blue balls is to distribute them equally among 18 identical cardboard boxes. There are 6 other ways in which all the blue balls can be distributed equally among identical cardboard boxes that are greater than 1; the number of identical cardboard boxes used is different in each of these ways. If the kindergarten manager also wants to distribute the red balls equally among the identical cardboard boxes previously used only to store the blue balls such that no red ball is left out, what is the minimum total number of the balls that a cardboard box will store?a)7b)10c)18d)30e)Cannot be DeterminedCorrect answer is option 'B'. Can you explain this answer?.

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