Question

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*}

Answer 1

hello :
the system of congruences is :
n≡ 2 ( mod 11)
n ≡ 3 (mod 17)
n = 11k+2......k ∈ N ....(*)
n = 17L +3......L ∈ 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)