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smc

number theory senior

Problem

When the natural numbers and , with , are divided by the natural number , the remainders are and , respectively. When and are divided by , the remainders are and , respectively. Then:
(A)
(B)
(C)
(D)
(E)
Solution
The divisors are the same, so take each variable modulo . That means Thus, and have the same remainder when divided by , so the answer is .
Final answer
E