Note: for Greatest common divisor . you can use the previous functions to help in prove it.

Prove the following functions are primitive recursive: . Quotient of two numbers: ifa, y N then quo(x, y)-z such that x-y * s+ r where . Reinainder of two numbers: if x, y N then rern(x,y)-r such that x y * + r . Divisibility function: if x, y E N then dio(x,y) 1 F3r E N such that y = r* x and where 0