We may use heat and trial method to solve.
For N = 1 , denomitor becomes larger then numerator
Same for N = 2.
Hence it's result is not an integer.
May be for larger N it divides, becuause for large N upper part , will give large result
If we multiply number and denomination by ( N + 1 ) then it becomes : ( (2 * N ) ! * / ( N ! * ( N + 1 ) ! )) *( (N +1 ) /2 ^(N + 1 ) )
where first term : ( (2 * N ) ! * / ( N ! * ( N + 1 ) ! )) is catalan number : https://en.wikipedia.org/wiki/Catalan_number
Hence it will reduces to Nth Catalan number * ( (N+1) / 2^(N+1) ) where catalan numbers will grow with large factor as N will increase as compared to power of 2 , and it may divide and result will be integer for some N
But overall if for any N it can't divide then answer will be not !
This type of problem are mostly based on observation, it's not necessary to do proof and all , as in gate exam speed matters, i will go for heat and trial first ( it saves time )