Solution
= Solution
A statistic $T$ is sufficient when the conditional law of the sample given $T$ is independent of the parameter. The factorisation criterion says this holds exactly when
$$
f(x;\theta)=g(T(x);\theta)h(x).
$$
If factorisation holds, cancellation in the conditional mass proves parameter independence. Conversely, multiply the parameter-free conditional mass given $T=t$ by the mass of $T=t$ to obtain the factorisation.