I am having trouble seeing why $r_k \equiv 0 (\mod A_i)$ for all $i \neq k$. Showing $b_k \equiv r_k (\mod A_k)$ was pretty trivial, but the other claim is giving me trouble. I could use some help.
2026-03-31 15:18:38.1774970318
Hungerford's Proof of the Chinese Remainder Theorem
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