How to solve a congruence system? Here is a tricky congruence system to solve, I have tried to use the Chinese Remainder Theorem without success so far.
$$2x \equiv3\;(mod\;7)\\
x\equiv8\;(mod\;15)$$
Thank you very much
Li

$$ 2x = 3(mod 7)    $$
$$ x+x= 3(mod 7)    $$
$$ x =  5(mod 7)    $$
$$ x =  8(mod 15)   $$
$$ x = 15k_1 + 8    $$
$$ x = 7k_2  + 5    $$
$$ x = 105k_3 + c   $$
$$ 0 \le c < 7*15   $$
$$ c(mod 7)=5       $$
$$ c = 15k_4+8      $$
$$ 0 \le k_4 < 7    $$
$$(15k_4+8)(mod 7)=3$$
$$ k_4 + 1 = 5      $$
$$ k_4 = 4          $$
$$ c=15*4+4 => c=68 $$
$$ x = 105k_3 + 68  $$
$$ 0 \le k_3 <\infty$$
$$ k_3 \in Z  $$
