\(\displaystyle{x}\equiv{1}\pm{o}{d}{\left\lbrace{4}\right\rbrace}\)

\(\displaystyle{x}\equiv{8}\pm{o}{d}{\left\lbrace{9}\right\rbrace}\)

\(\displaystyle{x}\equiv{10}\pm{o}{d}{\left\lbrace{25}\right\rbrace}\)

First let us solve the first two congruences.

\(\displaystyle{x}\equiv{1}\pm{o}{d}{\left\lbrace{4}\right\rbrace}\)

\(\displaystyle{x}\equiv{8}\pm{o}{d}{\left\lbrace{9}\right\rbrace}\)

We see that the solution x is unique modulo \(4.9=36\).

Now, \(9-4(2)=1\).

Thus,

\(x=1.9-8.4(2)\)

\(x=9-64\)

\(x=-55\)

\(\displaystyle{x}=-{19}\pm{o}{d}{\left\lbrace{36}\right\rbrace}\)

\(\displaystyle{x}={17}\pm{o}{d}{\left\lbrace{36}\right\rbrace}\)

Therefore, \(\displaystyle{x}={17}\pm{o}{d}{\left\lbrace{36}\right\rbrace}\).

Now, let us solve the below equations.

\(\displaystyle{x}\equiv{17}\pm{o}{d}{\left\lbrace{36}\right\rbrace}\)

\(\displaystyle{x}\equiv{10}\pm{o}{d}{\left\lbrace{25}\right\rbrace}\)

We see that the solution x is unique modulo \(36.25=900\).

Now, \(25(13)-36(9)=1\).

Thus,

\(x=17.25(13)-10.36(9)\)

\(x=5525-3240\)

\(x=2285\)

\(\displaystyle{x}={485}\pm{o}{d}{\left\lbrace{900}\right\rbrace}\)

Therefore, \(x = 485.\)