f: ℤ675,683 → ℤ821 × ℤ823 satisfies f(x) = (mod821(x), mod823 (x)). You may use: 1=411⋅821− 410⋅823 = 337,431 − 337,430 (*) (a) Give a similar definition for f^−1: ℤ821 × ℤ823 → ℤ675,683 . (b) Which integer x ∈ ℤ satisfies all 3 of the following conditions: (i) 1 ≤ x ≤ 675,683; (ii) x≡135 (mod 821); and (iii) x≡3 (mod 823)?

f: ℤ675,683 → ℤ821 × ℤ823 satisfies f(x) = (mod821(x), mod823 (x)). You may
use:
1=411⋅821− 410⋅823 = 337,431 − 337,430 (*)
(a) Give a similar definition for f^−1: ℤ821 × ℤ823 → ℤ675,683 .
(b) Which integer x ∈ ℤ satisfies all 3 of the following conditions:
(i) 1 ≤ x ≤ 675,683; (ii) x≡135 (mod 821); and (iii) x≡3 (mod 823)?