Binary operation - finding identity

Question

Here is the question:

The binary operation * is defined for $x,y \in\ S = {0,1,2,3,4,5,6}$ by

$x*y=(x^3y-xy)mod7$

Find the element $e$ such that $e*y=y$ for all $y\in\ S$

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So far I have the following:

Suppose $e * y=y$

$e * y = (e^3y-ey)mod7 = y$

$e^3y-ey=ymod7$

I don't know how to go from here... I need to figure out the value of $e$ - apparently it is 5.

Help would be greatly appreciated! Thanks!

Answer 1

$$e * y = (e^3y-ey)\bmod7 = y$$ Thus, $$e^3y - ey = 7k + y$$ for some $k \in \mathbb Z$ $$e^3y - ey - y = 7k$$ $$y(e^3-e-1) = 7k$$

The $LHS$ must be divisible by $7$ for all $y \in S$. Thus, $$(e^3 -e -1)\equiv 0 \pmod 7$$

Here, as $e \in S$, it is easy to check using trial and error that $e=5$.