(a) Define r(n, m) : N x Z+ ->N be the remainder obtained when dividing m into n. Define the function g: Z+ x Z+ ->Z+ as follows: g(a, b) = b if r(a, b) = 0, and g(a, b) = g(b, r(a,b)) otherwise. Describe what g is calculating, and justify your answer.

There is two cases:

first case: $g(a,b)=a\cdot\frac{b}{a}=b$

function multiplies two positive integers (because r(a, b) = 0); for example g(2,4)=2x2=4, g(3,9)=3x3=9, g(3,15)=3x5=15

second case - the result of function is remainder of dividing two positive integers; for example: g(7,8)=1, g(5,19)=4