Solve the Congruency

Computer Science Level 4

Given integers \(R,M\) with \(M\neq 0\), let \(S(R,M)\) denote the smallest positive integer \(x\) satisfying the congruence \[Rx \equiv 1 \pmod{M}\] if such an \(x\) exists. If such an \(x\) does not exist, put \(S(R,M)=0\).

Each line of this text file contains a pair of space separated integers representing \(R\) and \(M\), respectively.

Let L be the list of integers whose \(k\)-th element is the value of \(S(R,M)\), where \(R\) and \(M\) are taken from the \(k\)-th line of the text file.

Let T be the sum of all elements of L. What are the last three digits of T?


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