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)
