Hey Jacky J!
Assuming that these n random variables are indexed by some parameter θ which is the maximum value in the distribution, then it's actually very easy to show. The Factorization Theorem says that if we can factor the joint density (or Likelihood function, L(x)) into a function of the data and a function of the parameter which relies only on the data only through a statistic T(x) as follows:
L(x) = h(x)g(T(x),θ)
then T(x) is sufficient for θ.
Given that F is the cdf, we usually write F' = f is the pdf, and so the joint likelihood L being the product of the n copies of the shared pdf can be written as follows:
L(x) = Πf(xi)*I{xi≤θ}
where I{.} is an indicator function which assumes a 1 if the statement inside is true and a 0 otherwise.
This likelihood evaluates to 0 whenever any of the xi is bigger than the maximum allowable value θ. However, because the maximum of the sample (x(n)) is the biggest observed value of x, we can rewrite the likelihood as:
L(x) = Πf(xi)*I{x(n)≤θ}
Choosing h(x) = Πf(xi) and T(x) = x(n), we find g(T(x),θ) = I{x(n)≤θ} and we have successfully factored the likelihood as the Factorization Theorem requires.
Therefore, x(n) (the sample maximum) is sufficient for θ
