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Sum of remainder pairs greater than 2017=N

Source: BdMO 2022 Secondary P6

April 12, 2022
number theory

Problem Statement

About 55 years ago, Joydip was researching on the number 20172017. He understood that 20172017 is a prime number. Then he took two integers a,ba,b such that 0<a,b<20170<a,b <2017 and a+b≠2017.a+b\neq 2017. He created two sequences A1,A2,…,A2016A_1,A_2,\dots ,A_{2016} and B1,B2,…,B2016B_1,B_2,\dots, B_{2016} where AkA_k is the remainder upon dividing akak by 20172017, and BkB_k is the remainder upon dividing bkbk by 2017.2017. Among the numbers A1+B1,A2+B2,…A2016+B2016A_1+B_1,A_2+B_2,\dots A_{2016}+B_{2016} count of those that are greater than 20172017 is NN. Prove that N=1008.N=1008.
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