Typical remainders

We know that $X=1990=2\times{5}\times{199}$ . Using Euler's Theorem, we have

$X\equiv\begin{cases}0\pmod{2}\\4\pmod{5}\\29\pmod{199}\end{cases}$

We can write those congruences into equations for $~t, s, \text{and}~~u$ :

$X=\begin{cases}2t\\5s+4\\199u+29\end{cases}$

Start by substituting firs equation to second congruence:

$2t\equiv4\pmod{5}\\t\equiv2\pmod{5}$

meaning that $t=5s+2$ . Substitute $t$ into the first equation:

$X=2t=10s+4$ . Substitute this $X$ into the third congruence:

$\begin{array}{c}&10s+4&\equiv29\pmod{199}\\2s&\equiv5\pmod{199}\\2s&\equiv204\pmod{199}\\s&\equiv102\pmod{199}\end{array}$

meaning that $s=199u+102$ . Finally,

$X=10(199u+102)+4=1990u+1024\equiv1024\pmod{1990}$