Discrete math Problem!

Question

Suppose f: integer(z) to integer(z) is a function with the property that f(a+b) = f(a) + f(b) for every two integers a and b. Prove that if f(c) is even for some odd integer c then f(x) is even for ever integer x.

Answer 1

I'd let c=a+b.  Then notice f(c) = f(a+b) = f(a) + f(b)

 

You know c is odd.   What does this tell you about the parity of a and of b?   [Hint: can they both be even? ]

You know f(c) is even.  What does that tell you about f(a) and f(b) ?    

 

Think about how parities combine:  odd+odd=even, odd+even=odd, even+even=even