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to get 4x ≡ 6 (mod 10) ??

you can divide by \(2\) again if you want to

thats right, notice that
\[ab \mid ac \implies b\mid c\]

\(2*10\mid 2(4x-6) \implies 10\mid (4x-6)\)

gcd(8,20)*

you did you get 6/4 ≡ 6/(-1)? Looks like you did 4 - 5 in the denominator?

our goal is to convert that fraction into an integer
4 is same as -1 in mod 5, so...

Maybe I'll stick to guess and check for now. I will study this method further

thank you @ganeshie8 :')