Pick any number, multiply the number by 3, add 6 to the product, divide the sum by 3, and subtract 2 from the quotient.

(a) What is the relationship between the number you started with and the final number?

(b) Arbitrarily select some different numbers and repeat the process, recording the original number and the result. 

(c) Can you make a conjecture about the relationship between the original number and the final number?

(d) Prove, using deductive reasoning, the conjecture you made in part (c).