Question 38
Determine the positive integers $a$, $b$, $c$ with the following properties:
(i) $(a^2 + b^2)(c^2 + 2023^2) = (ab + 2023c)^2$;
(ii) $(a^2 + 2023^2)(b^2 + c^2) = (2023a + bc)^2$;
(iii) the greatest common divisor of $a$, $b$, $c$ and $2023$ equals $1$.
(i) $(a^2 + b^2)(c^2 + 2023^2) = (ab + 2023c)^2$;
(ii) $(a^2 + 2023^2)(b^2 + c^2) = (2023a + bc)^2$;
(iii) the greatest common divisor of $a$, $b$, $c$ and $2023$ equals $1$.