- #1

Bashyboy

- 1,421

- 5

## Homework Statement

Let ##n## be some natural number. Solve the following ##n \times n## homogeneous system of equations:

$$\sum_{1|i} x_i = 0$$

$$\sum_{2|i} x_i = 0$$

$$\vdots$$

$$\sum_{n|i} x_i = 0$$,

where ##a|b## means ##b## is divisible by ##a##.

## Homework Equations

## The Attempt at a Solution

After working through examples, it seems that ##x_i = 0## holds for all ##i \in \{1,...,n\}##. Since ##1|i## holds for ##i=1,2,...,n##, the first equation becomes ##\sum_{i=1}^n x_i = 0##. Moreover, since ##n|i## happens if and only if ##i=n##, the last equation reduces to ##x_n = 0##. Through my experiments I noticed that the "second" half of the system equations reduced to a single variable equal to ##0##; i.e., ##x_i = 0## for ##i \in \{n - [\frac{n}{2}],...,n\}##, where ##[a]## denote the greatest integer function.

I tried several times to use these observations in solving the problem, but I didn't have much luck. I could use a hint.