Below is a complete algorithmic solution (i.e. nothing is pulled out of a hat) using Easy CRT. Generally this method is faster than hoping for lucky guesses (it took me $20$ seconds mentally).
If $\rm\:\ x \equiv \color{brown}2\ mod\ 25,\ \color{green}7\ mod\ 11, \ \color{red}1\ mod\ 8,\ $ then applying Easy CRT we quickly compute
$\rm x \equiv \color{brown}2 + 25\:\!\left(\!\frac{\color{green}7\!-\!\color{brown}2}{25}\ mod\ 11\!\right)\equiv -\color{blue}{48}\:\ (mod\ 25\,\!\cdot\! 11)\ \ \ by\ \ \ mod\ 11\!:\, \frac{5}{25}\equiv \frac{1}5\equiv\frac{-10}5\equiv -2 $
$\rm x\equiv\:\! -\color{blue}{48} + 25\,\!\cdot\! 11\:\!\left(\!\frac{\color{red}1\!+\!\color{blue}{48}}{25\,\!\cdot\! 11}\ mod\ 8\!\right)\equiv 777\:\ (mod\ 25\,\!\cdot\!11\!\cdot\! 8)\ \ \ by\ \ \ mod\ 8\!:\, \frac{49}{3\,\!\cdot\! 1}\equiv \frac{9}3\equiv 3\ \ $