<s>[INST] system of modular equations. $x\equiv 2\pmod3$
$x\equiv 3\pmod 5$
$x\equiv 7 \pmod{11}$
How can I solve this system for $x$? I've tried all kinds of things using divisibility but no success. Any hints of solutions are greatly appreciated. What value of $x$ satisfies these three equations?
 [/INST] Here is a hands-on solution:
$x\equiv 2\pmod3$ implies $x=2+3y$.
$x\equiv 3\pmod 5$ then implies $3y\equiv 1 \pmod 5$ and so $y\equiv 2 \pmod 5$ because $2 \cdot 3 \equiv 1 \pmod 5$. This means that $y=2+5z$ and $x=2+6+15z=8+15z$.
$x\equiv 7 \pmod{11}$ then implies $15z \equiv -1 \pmod{11}$ and so $z \equiv -3 \pmod{11}$ because $15 \equiv 4  \pmod{11}$ and $3 \cdot 4 \equiv 1  \pmod{11}$. This means that $z=-3+11t$ and so $x=8-45+165t=-37+165t$.
Thus, the solution is $x \equiv -37 \equiv 128 \pmod{165}$.
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