<s>[INST] Finding a polynomial mod 5: What did they do here? I'm reading through this solution for the problem below.

The given solution is this:

What I don't quite understand is how they got say from $\frac{1}{3}(X-2)(X-4)$ to be 'equal' to $2(X-2)(X-4)$.
How did they go from one step to the other? I am not quite understanding this.
 [/INST] All my arithmetic is modulo 5:
$5(X-2)(X-4)=0$
Any of ${1,2,3,4}$ when multiplied by $3$ is non-zero, so
$\frac{5}{3} (X-2)(X-4)=0 \Rightarrow$
$ \frac{6}{3} (X-2)(X-4)- \frac{1}{3} (X-2)(X-4)=0$
Hence $ \frac{1}{3} (X-2)(X-4)= 2(X-2)(X-4)$
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