Question $2$ Assume that we have unlimited supply of coins of each denomination. For each of the following cases does the change making problem by the

greedy algorithm always find an optimal solution? Justify your answer.

a$-)$ Each coin in the series is worth at least twice the next lower denomination, and the series includes a $1-$unit coin.

b$-)$ The available denominations are $p^k,p^{k-1},$cdots,$p,1$ where $p>1$ and $k0.$