Imagine that a new coin that is worth exactly 12 cents has been introduced to our existing currency system. Prove or disprove the statements (use an example or counterexample):

(a) The cashiers algorithm using quarters, dimes, nickels, 12-cent coins, and pennies can produce coins change using fewer coins than the algorithm without the 12 cent coin.

(b) The cashiers algorithm using quarters, dimes, nickels, 12-cent coins, and pennies and will produce change using the fewest coins possible for all coin values.