Question

To determine: The smallest nonnegative integer x that satisfies the given system of congruences. x\equiv 3\pmod 5 x\equiv 7\pmod 8

Congruence
ANSWERED
asked 2021-04-16
To determine: The smallest nonnegative integer x that satisfies the given system of congruences.
\(\displaystyle{x}\equiv{3}\pm{o}{d}{5}\)
\(\displaystyle{x}\equiv{7}\pm{o}{d}{8}\)

Answers (1)

2021-04-18
\(\displaystyle{x}\equiv{3}\pm{o}{d}{5}\)
\(\displaystyle{x}\equiv{7}\pm{o}{d}{8}\)
We see that the solution x is unique modulo 5.8=40.
Now, 8(2)-5(3)=1.
Thus,
x=3.8(2)-7.5(3)
x=48-105
x=-57
\(\displaystyle{x}=-{17}\pm{o}{d}{\left\lbrace{40}\right\rbrace}\)
\(\displaystyle{x}={23}\pm{o}{d}{\left\lbrace{40}\right\rbrace}\)
Therefore, x =23.
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