AIME 2026a Problem 13

For each nonnegative integer $r$ less than $502$, define$$S_r=\sum_{m\geq 0}\binom{10,000}{502m+r},$$where $\binom{10,000}{n}$ is defined to be $0$ when $n>10,000$. That is, $S_r$ is the sum of all the binomial coefficients of the form $\binom{10,000}{k}$ for which $0\leq k\leq 10,000$ and $k-r$ is a multiple of $502$. Find the number of integers in the list $S_0,S_1,S_2,\dots,S_{501}$ that are multiples of the prime number $503$.