Professor Elmtwig makes a monthly payment of $$1600$ on his mortgage. The original term of the mortgage is for $30$ years at $3\mathrm{.}75\%$ interest compounded semi$-$annually. His lender suggests that if he makes bi$-$weekly payments of $$800,$ it will reduce how long it will take to pay off the mortgage. Answer the following questions for $3$ points each.

a$)$ How much did Professor Elmtwig originally borrow?

b$)$ How much faster will the mortgage be paid off if Elmtwig makes the bi$-$weekly payments $(26$ payments per year$)?$ HINT: You may use the formula to solve for N$,$ but the TVM functions on your calculator will be much faster. Express your answer in terms of years $($to two decimal places$).$

c$)$ Is paying $$800$ bi$-$weekly the same as paying $$1600$ monthly? Why or why not? $($HINT: How many payments per year?$)$