Question

# To determine: The smallest nonnegative integer x that satisfies the given system of congruences. x\equiv 6\pmod 8 x\equiv 17\pmod {25}

Congruence
To determine: The smallest nonnegative integer x that satisfies the given system of congruences.
$$\displaystyle{x}\equiv{6}\pm{o}{d}{8}$$
$$\displaystyle{x}\equiv{17}\pm{o}{d}{\left\lbrace{25}\right\rbrace}$$

2021-04-21
$$\displaystyle{x}\equiv{6}\pm{o}{d}{8}$$
$$\displaystyle{x}\equiv{17}\pm{o}{d}{\left\lbrace{25}\right\rbrace}$$
We see that the solution x is unique modulo 25.8=200.
Now, 25(1)-8(3)=1.
Thus,
x=6.25(1)-17.8(3)
x=150-408
x=-258
$$\displaystyle{x}=-{58}\pm{o}{d}{\left\lbrace{200}\right\rbrace}$$
$$\displaystyle{x}={142}\pm{o}{d}{\left\lbrace{40}\right\rbrace}$$
Therefore, x=142.