Gamma function of positive integer?

Questions
1

Is it possible to write that
Gamma[n]=(n-1)! if n is a positive integer at Mathematica?
Explicitly, when i execute the code

LaplaceTransform[t^n,t,s]

it produces
s^(-1-n)Gamma(n+1).
But,when n is an integer i want Mathematica to produce
s^(-1-n).n!

Dear Friends, Thanks a lot. Have a good study.

2

Try:

FullSimplify[LaplaceTransform[t^n, t, s], Assumptions -> n \[Element] PositiveIntegers]
3

I am very thankful to you.