Question 87
The elements of the set $\{1,2,3,4,5,6,7,8,9,10,11\}$ can be separated in the sets $\{3,9,10,11\}$ and $\{1,2,4,5,6,7,8\}$ so that each of them has the sum of the elements equal to 33.
a. Display a way to separate the elements of the set $\{1,2,3,4,5,6,7,8\}$ into three sets such that the sums of the elements of each set are the same.
b. Explain why it is not possible to separate the numbers from the set $\{1,2,3,4,5,6,7,8,9,10\}$ into two sets of the same sum.
c. For each positive integer $n \geq 2$, determine the smallest positive integer $N$ such that the set $\{1,2, \ldots, N\}$ can be separated into exactly $n$ sets of the same sum.
a. Display a way to separate the elements of the set $\{1,2,3,4,5,6,7,8\}$ into three sets such that the sums of the elements of each set are the same.
b. Explain why it is not possible to separate the numbers from the set $\{1,2,3,4,5,6,7,8,9,10\}$ into two sets of the same sum.
c. For each positive integer $n \geq 2$, determine the smallest positive integer $N$ such that the set $\{1,2, \ldots, N\}$ can be separated into exactly $n$ sets of the same sum.