$3.$Patrick received a mortgage for a property he was interested in investing in for a while. The mortgage lasts thirty years with a $6\mathrm{.}75\%$ annual interest rate for a loan of $$500,000.$ Calculate the interest and principal payments for the first six years of the mortgage. $($Show your calculations$).$ What will be the remaining principal balance at the end of the $17$th and $27$th years? $($You only need to specify these two numbers$).$