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
ANSWERED
asked 2021-05-04
To determine: The smallest nonnegative integer x that satisfies the given system of congruences.
\(\displaystyle{x}\equiv{3}\pm{o}{d}{\left\lbrace{1917}\right\rbrace}\)
\(\displaystyle{x}\equiv{75}\pm{o}{d}{\left\lbrace{385}\right\rbrace}\)

Answers (1)

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.
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