Divisible By .....

The fact holds $\forall n\in\mathbb N_0$ , and it should be specified.

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Proof by induction:

For $n=0$ , the expression is equal to $9$ . Assume $k$ satisfies the given fact. Then we know:

5^{2n}\equiv 6n-8\equiv 6n+1\pmod 9\tag{1}

Now let us prove $k+1$ satisfies the fact:

$5^{2(n+1)}-6(n+1)+8= 25\cdot 5^{2n}-6n+2\stackrel{(1)}\equiv 25\cdot (6n+1) - 6n +2$

$\equiv 7(6n+1)+3n+2\equiv 42n+7+3n+2\equiv 45n+9\equiv 0\pmod 9\square$

put value of n =1 then its 5^{2} - 6 + 8 = 27 which is divisible by 9....