Putnam 1994 B6

For any integer $n$, set $$n_a = 101a - 100\cdot 2^a.$$ Show that for $0 \leq a,b,c,d \leq 99$, $n_a + n_b \equiv n_c + n_d \pmod{10100}$ implies $\{a,b\} = \{c,d\}$.