HMMT November 2022 U33

A group of 101 Dalmathians participate in an election, where they each vote independently on either candidate $A$ or $B$ with equal probability. If $X$ Dalmathians voted for the winning candidate, the expected value of $X^{2}$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $\operatorname{gcd}(a, b)=1$. Find the unique positive integer $k \leq 103$ such that $103 \mid a-b k$.