When doing a problem on computer arithmetic with large integers, I reached a step when I needed to solve system of congruences. I came up with the following equations: $$x ? 65 pmod{99})\ x ? 2 pmod{98}\ x ? 51 pmod{97}\ x ? 10 pmod{95}$$ I tried to solve using chinese remainder theorem, but it became too large and complex. The original question is : Find sum of numbers $123,684$ and $413,456$ by representing the numbers as $4$-tuple by using remainder modulo of pair-wise relatively prime numbers less than $100.$ While solving by chinese remainder theorem the inverse modulo becomes...