  With help of $q$-congruence, we prove the divisibility of some binomial sums.
For example, for any integers $\rho,n\geq 2$, $$\sum_{k=0}^{n-1}(4k+1)
\binom{2k}{k}^\rho \cdot (-4)^{\rho(n-1-k)} \equiv
0\pmod{2^{\rho-2}n\binom{2n}{n}}.$$
