Simultaneous Solution for Two Modular Equations with Known Integers

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The discussion focuses on solving two modular equations: x % n1 = 0 and x % n2 = 1, where n1 and n2 are known positive integers. The Chinese Remainder Theorem (CRT) guarantees the existence of a solution for these equations. The conversation explores efficient methods to find the simultaneous solution for x without resorting to trial and error. Techniques discussed include leveraging the properties of modular arithmetic to derive a direct formula for x.

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mohamed el teir
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let equation 1: x % n1 = 0, equation 2: x % n2 =1, where n1 and n2 are known positive integers, any multiple of n1 will solve eqn1 and any multiple of n2 (and adding 1 to the multiple) will solve eqn2, but is there a short way to simultaneously solve the two equations to find x instead of continuously trying to find the first multiple of n2 that if 1 is added to it will be a multiple of n1 ?
 
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