Chinese Remainder Theorem problem 7

Question

I have been set the task of solving the following set of congruences using CRT.

$$\begin{align} x&=2\mod 11, \\ x&=3 \mod 12, \\x&=4 \mod 15.\end{align}$$

I solved the equation of $M_1 = 12 \times 15 = 180$ and then $1 \equiv 180x \mod 11$ to get $x=3$.

My next equation to solve would be $M_2 = 11 \times 15 =165$ and then $1 \equiv 165x \mod 12$ however this cannot be solved so I'm wondering where I have gone wrong.

This is my second time attempting a CRT question after being shown the method above in my lecture however I cannot seem to see what I am doing wrong.

Thanks for your help!

Answer 1

I did not go into details about the computation, but just notice that $12$ and $15$ have a common factor $3$. So in some cases there will not be any solutions (the CRT does not apply here).