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
To determine: The smallest nonnegative integer x that satisfies the given system of congruences.
$$\displaystyle{x}\equiv{3}\pm{o}{d}{5}$$
$$\displaystyle{x}\equiv{7}\pm{o}{d}{8}$$

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.