Find the smallest positive integer that satisfies the system of congruences \begin{align*} n & \equiv 2 \pmod{11}, \\ n & \equiv 3 \pmod{17}. \end{align*}

Hello : 
the system of congruences is : 
n≡ 2 ( mod 11)
n ≡ 3 (mod 17) 
n = 11k+2k ∈ N    (*)
n = 17L +3L ∈ N
17L +3 = 11k+2
11k = 17L +1(1) 
by (1) : 11k ≡ 1  (mod 17)
33k ≡ 3(mod 17)...(2)
but : 33 ≡ -1 (mod 17) and  -3 ≡  14 (mod 17)
(2) : - k ≡3 (mod 17)
k≡ -3 (mod 17)  
k≡ 14 (mod 17) 
k = 17a+14
subsct in (*) :  n = 11(17a+14)+2
all positive integer that satisfies the system is : n = 187a +156...   a ∈ N 
all  smallest integer that satisfies the system is : n = 187+156 = 343 (when : a=1)