Question 13
For each nonnegative integer $r$ less than $502$ define $$
S_r = \sum_{m \ge 0} \binom{10{,}000}{502m + r},
$$ where $\binom{10{,}000}{n}$ is defined to be $0$ when $n \gt 10{,}000.$ That is, $S_r$ is the sum of all the binomial coefficients of the form $\binom{10{,}000}{k}$ for which $0 \le k \le 10{,}000$ and $k - r$ is a multiple of $502.$
Find the number of integers in the list $S_0,$ $S_1,$ $S_2,$ $\ldots,$ $S_{501}$ that are multiples of the prime number $503.$
S_r = \sum_{m \ge 0} \binom{10{,}000}{502m + r},
$$ where $\binom{10{,}000}{n}$ is defined to be $0$ when $n \gt 10{,}000.$ That is, $S_r$ is the sum of all the binomial coefficients of the form $\binom{10{,}000}{k}$ for which $0 \le k \le 10{,}000$ and $k - r$ is a multiple of $502.$
Find the number of integers in the list $S_0,$ $S_1,$ $S_2,$ $\ldots,$ $S_{501}$ that are multiples of the prime number $503.$