Question

prove to disprove:
a)If there exists an integer solution to ax^2+by^2=c, then gcd(a,b)|c
b)if gcd(a,b)|c, there exists an integer solution to ax^2+by^2=c.
I don't know where to start..any help please.. thank you~

Answer 1

a) first part is clear, suppose there exists an integer solution to ax^2+by^2=c and\[d=\gcd(a,b)\]then\[d|ax^2+by^2=c \ \ \rightarrow \ \ d|c\]

Answer 2

b) second part, let a=1, b=2 and c=3...see what happens