subject Number theory 3rd year course Thank you for your hel

subject: Number theory (3rd year course)

Thank you for your help!

Solution

Case1: a, b, both odd.
Then ab is odd.
so f(ab) = f(odd) = 1.
On the other hand, f(a) = 1 = f(b), so f(a)f(b) = 1*1 = 1 = f(ab).

Case 2:a odd, b even (same as the case when a is even and b odd)
ab is even.
0 = f(even) = f(ab) = f(a)f(b) = f(odd)f(even) = 1*0 = 0.

Case .3:a,b both even. then ab is even
0 = f(even) = f(ab) = f(a)f(b) = f(even)f(even) = 0*0 = 0

If (a,b) = 1 means that they are relatively prime.

f(ab)=f(a)*f(b) is multiplicative function