Explanation:

Given that,
Number $$517x324$$ is divisible by $$12$$
So, the multiple of $$12=3\times 4$$
The mentioned number will be divisible by 12 if it is individually divisible by $$3$$ and $$4$$
Divisibility by $$4$$- Any number will be divisible by $$4$$ if last two digit of that number will be divisible by $$4$$.
so, in this case last two digits are $$24$$, which is divisible by $$4$$.
Hence, for any value of $$x$$ the mentioned number will be divisible by $$4$$
Divisibility by $$3$$- Any number will be divisible by $$3$$, if the sum of the digit of that number will be divisible by $$3$$.
But as per question we have to find the minimum value of $$x$$, so that i can be divisible.
Hence, $$5+1+7+x+3+2+4=22+x$$
the nearest value of$$22+x$$ which is divisible by $$3=24$$ ,
So $$22+x=24$$
$$x=2$$