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+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)