A couple decide to borrow $$250000$ to buy a house. Interest is calculated monthly on the balance still owing, at a rate of $6\mathrm{.}06\%$ per annum. The loan is to be repaid at the end of $15$ years with equal monthly repayments of $$M.$ Let $$A_n$ be the amount owing after the $n^{th}$ repayment.

$($a$)$ Derive an expression for $A_{60}.$

$($b$)$ Find the value of $M.$

$($c$)$ Hence, calculate the amount still owing after $5$ years of payment at this rate.

$($d$)$ At the end of five years, the interest rate is increased to $7\mathrm{.}2\%$ per annum and the couple change their payments to $$1800$ per month. How many more months are needed to pay off the remainder of the loan