There are 48 balls.
These balls are put into five boxes so that it may be applied to the following conditions 1 and 2.
<Condition 1>
Five or more balls are put into every box.
<Condition 2>
As for every two boxes, the common divisor of the number of ball in each box is 1 only.
Find all groups of the each number of balls in five boxes.
These balls are put into five boxes so that it may be applied to the following conditions 1 and 2.
<Condition 1>
Five or more balls are put into every box.
<Condition 2>
As for every two boxes, the common divisor of the number of ball in each box is 1 only.
Find all groups of the each number of balls in five boxes.
Answer
Solution
Since 6 is the multiple of 2 and 3, it is only a prime number that put into the last two boxes.
The group of two prime numbers with 30 of the sum is 11 + 19 and 13 + 17.
② In case first two numbers are 5 and 6, the next candidate numbers are 8, 9, 10.
However these numbers are multiple of 2 and/or 3, these are unsuitable.
The next candidate is 11 and the following prime number is 13 according to 48 - (5 + 6 + 11) = 25.
There is no applied number based on 25 - 13 = 12.
Therefore, there is no combination of numbers in case first two numbers are 5 and 6.
③ In case first two numbers are 5 and 7, the next candidate number is 8.
In the group of two numbers with the sum of 28, the group of (multiple of 3) + (prime number) or (prime number) + (prime number) should be found.
There are 2 groups, 9+19 and 11+17.
④ In case first two numbers are 5 and 7, the next candidate number is 9.
In the group of two numbers with the sum of 27, the group of (multiple of 2 with no multiple of 5) + (prime number) or (prime number) + (prime number) should be found.
There is one group, 11+16.
⑤ In case first two numbers are 5 and 7, the next candidate number is 11.
In the group of two numbers with the sum of 25, the group of (multiple of 2 with no multiple of 5) + (prime number) or (prime number) + (prime number) should be found.
There is one group, 12+13.
⑥ In case first two numbers are 5 and 7, the next candidate number is 13.
Since the 4th number should be bigger than 13, there is no suitable group of number.
There is no combination of numbers in case first two numbers are 5 and 7.
⑦ In case first two numbers are 5 and 8, the next candidate number is 9.
Since the 4th number should be 13 or more, the sum total cannot be set to 24 by two.
There is no suitable group of number.
There is no combination of numbers in case first two numbers are 5 and 8.
⑧ In case first two numbers are 5 and 9, there is no combination of numbers according to the same reason as ⑦.
⑨ In case first two numbers are 6 and 7, the next candidate number is 11.
This is same as ⑦ and there is no combination of numbers in case first two numbers are 6 and 7.
⑩ In case first two numbers are 7 and 8, the next candidate number is 9.
In the group of two numbers with the sum of 24, the group of (prime number) + (prime number) should be found.
There is one group, 11+13.
⑪ As for the group of 7, 8 and 11 or more, there is no combination of numbers can not be found.
Therefore there are seven groups in total to be found.
Solution
According to the condition 2, the number of the balls which put into five boxes is three kinds of numbers which are multiple of 2, multiple of 3, and a prime number.
① In case first two numbers are 5 and 6, the next candidate number is 7.
① In case first two numbers are 5 and 6, the next candidate number is 7.
Since 6 is the multiple of 2 and 3, it is only a prime number that put into the last two boxes.
48 - (5 + 6 + 7) = 30.
The group of two prime numbers with 30 of the sum is 11 + 19 and 13 + 17.
② In case first two numbers are 5 and 6, the next candidate numbers are 8, 9, 10.
However these numbers are multiple of 2 and/or 3, these are unsuitable.
The next candidate is 11 and the following prime number is 13 according to 48 - (5 + 6 + 11) = 25.
There is no applied number based on 25 - 13 = 12.
Therefore, there is no combination of numbers in case first two numbers are 5 and 6.
③ In case first two numbers are 5 and 7, the next candidate number is 8.
48 - (5 + 7 + 8) = 28.
In the group of two numbers with the sum of 28, the group of (multiple of 3) + (prime number) or (prime number) + (prime number) should be found.
There are 2 groups, 9+19 and 11+17.
④ In case first two numbers are 5 and 7, the next candidate number is 9.
48 - (5 + 7 + 9) = 27.
In the group of two numbers with the sum of 27, the group of (multiple of 2 with no multiple of 5) + (prime number) or (prime number) + (prime number) should be found.
There is one group, 11+16.
⑤ In case first two numbers are 5 and 7, the next candidate number is 11.
48 - (5 + 7 + 11) = 25.
In the group of two numbers with the sum of 25, the group of (multiple of 2 with no multiple of 5) + (prime number) or (prime number) + (prime number) should be found.
There is one group, 12+13.
⑥ In case first two numbers are 5 and 7, the next candidate number is 13.
48 - (5 + 7 + 13) = 23.
Since the 4th number should be bigger than 13, there is no suitable group of number.
There is no combination of numbers in case first two numbers are 5 and 7.
⑦ In case first two numbers are 5 and 8, the next candidate number is 9.
48 - (5 + 8 + 9) = 26.
There is no group of two prime numbers of 11 or more with the sum of 26.
The next candidate number is 11.
There is no group of two prime numbers of 11 or more with the sum of 26.
The next candidate number is 11.
48 - (5 + 8 + 11) = 24.
Since the 4th number should be 13 or more, the sum total cannot be set to 24 by two.
There is no suitable group of number.
There is no combination of numbers in case first two numbers are 5 and 8.
⑧ In case first two numbers are 5 and 9, there is no combination of numbers according to the same reason as ⑦.
⑨ In case first two numbers are 6 and 7, the next candidate number is 11.
48 - (6 + 7 + 11) = 24.
This is same as ⑦ and there is no combination of numbers in case first two numbers are 6 and 7.
⑩ In case first two numbers are 7 and 8, the next candidate number is 9.
48 - (7 + 8 + 9) = 24.
In the group of two numbers with the sum of 24, the group of (prime number) + (prime number) should be found.
There is one group, 11+13.
⑪ As for the group of 7, 8 and 11 or more, there is no combination of numbers can not be found.
Therefore there are seven groups in total to be found.
