用数学归纳法证明：对于所有正整数 $n$，$3^{6n}+92^n+89$ 都能被 $91$ 整除。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

按归纳法完整写出基础步骤、归纳假设和由k到k+1的递推。利用3的6次方等于728+1，且728是91的倍数。

Let $F(n)=3^{6n}+92^n+89$.

For $n=1$,

so the statement is true.

Assume that $F(k)$ is divisible by $91$ for some positive integer $k$. Since $3^6=729=728+1$ and $92=91+1$,

The first term is divisible by $91$ by the inductive hypothesis, and the second is explicitly a multiple of $91$. Hence $F(k+1)$ is divisible by $91$.

Therefore, by mathematical induction, $\boxed{3^{6n}+92^n+89\text{ is divisible by }91}$ for every positive integer $n$.