Question

# To determine: The smallest nonnegative integer x that satisfies the given system of congruences. x\equiv 3\pmod {1917} x\equiv 75\pmod {385}

Congruence
To determine: The smallest nonnegative integer x that satisfies the given system of congruences.
$$\displaystyle{x}\equiv{3}\pm{o}{d}{\left\lbrace{1917}\right\rbrace}$$
$$\displaystyle{x}\equiv{75}\pm{o}{d}{\left\lbrace{385}\right\rbrace}$$

2021-05-06
$$\displaystyle{x}\equiv{3}\pm{o}{d}{\left\lbrace{1917}\right\rbrace}$$
$$\displaystyle{x}\equiv{75}\pm{o}{d}{\left\lbrace{385}\right\rbrace}$$
We see that the solution x is unique modulo 385.1917=738045.
Now, 1917(48)-385(239)=1.
Thus,
x=75.1917(48)-3.385(239)
x=6901200-276045
x=6625155
$$\displaystyle{x}={720795}\pm{o}{d}{\left\lbrace{738045}\right\rbrace}$$
Therefore, x = 720795.