Children with torches puzzle 
It's almost nighttime and some children are trying to light up their backyard. They have $n$ torches ($n \in  \mathbb{N}$) and want to distribute themselves on the $n \text{ } \text{x} \text{ }n$-Meter backyard in such a way, that no two torches illuminate each other. Each torch sends light in 8 different directions, as in the following picture:

There are horizontal beams (marked in red), vertical beams (marked in green) and diagonal beams (marked in yellow).
We suppose that $n \geq 5$ and that $n$ is not divisible by $2$ nor $3$. Prove that the following positioning of $n$ children with torches $T_0, T_1, ..., T_{n-1}$ works, i.e no two torches light the same position in the backyard:
For $0 \leq i \leq n-1$ we position the torch $T_i$ on the field ($i, 2i \text{ } \text{mod } n).$
Here, we use the ($x$-coordinate, $y$-coordinate) coordinate system, where $x$ describes the horizontal position, and $y$ the vertical. For example: The three torches in the picture are placed on the fields $(3, 1), (2, n-3)$ and $(n-2, n-2).$

My idea was to prove by contradiction and break up each case on how the torches light up their path (horizontally, vertically and diagonally), but I can't see what follows. Can someone offer their thoughts or point me in the right direction?
 A: You need to show that...


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*No two torches have the same $x$-coordinate. This is obvious, as the $x$-coordinate of the $i^{th}$ torch is $i$.

*No two torches have have the same $y$-coordinate. This is not obvious. If the $i^{th}$ and $j^{th}$ torches had the same $y$ coordinate, it would mean $$2i\equiv 2j\pmod n$$Now, you can use some modular arithmetic to deduce this is only possible when $i\equiv j\pmod n$. (Hint: the equation above is equivalent to $2(i-j)\equiv 0\pmod n$. What is the definition of $\equiv\pmod n$?)



*

*No two torches are on he same upward sloping diagonal. You need to think about how the $(x,y)$ coordinates of a torch relate to the upward diagonal it is on. For example, the main diagonal is the one where $x=y$, and for the one above it, $y=x+1$. In other words, the diagonals consist of squares $(x,y)$ for which $y-x$ is constant. If the $i^{th}$ and $j^{th}$ torches were on the same diagonal, then it would then be the case that $$ 2i-i\equiv 2j-j\pmod n$$which quickly leads to a contradiction.





*

*For the downward sloping diagonals, two squares are now on the same diagonal if and only if the sum of their $x$ and $y$ coordinates is the same, so you instead would get $$i+2i\equiv j+2j\pmod n$$which leads to a contradiction in a similar manner to the second bullet point.

