Find the remainder of

$\large{\frac{18^7 + (7)(18^6) + (21)(18^5) +(35)(18^4) + (35)(18^3) + (21)(18^2) +7(18) +18}{19}}$ .

The answer is 17.

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Note that the expression is can be written as $\frac{(18+1)^7 + 17}{18+1}$ .

Thus,

$\Rightarrow (18+1)^6 + \frac{17}{19} \Rightarrow$ the remainder is $17$ .