Question 4
Determine the values of the positive integer $n$ for which the following system of equations has a solution in positive integers $x_1,x_2,\ldots,x_n$. Find all solutions for each such $n$.
$$x_1+x_2+\cdots+x_n=16\qquad(1)$$
$$\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_n}=1\qquad(2)$$
$$x_1+x_2+\cdots+x_n=16\qquad(1)$$
$$\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_n}=1\qquad(2)$$