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## Homework Statement

http://img200.imageshack.us/img200/7097/99175506.gif [Broken]

## Homework Equations

## The Attempt at a Solution

For [tex]n \in N[/tex] let P(n) be the statement: "81 | (10

^{n+1}-9n-10)"

**Base Case:**when n=1: 10

^{n+1}-9n-10 = 81 = 81 × 1

So P(1) is true

**Inductive Step:**let [tex]k \in N[/tex] and suppose [tex]P(k)[/tex] is true, that is

**81 | (10**is true. Then [tex]10^{k+1}-9k-10=81m[/tex] for some [tex]m \in Z[/tex]. Then [tex]10^{k+1}=(81m+9k+10)[/tex].

^{k+1}-9k-10)So,

**10**

=10 × (81m+9k+10)-10k+10-10

= 10 × (81m+9k+10-k)

^{k+2}-10(k+1)-10=10 × 10^{k+1}-10(k+1)-10=10 × (81m+9k+10)-10k+10-10

= 10 × (81m+9k+10-k)

I'm stuck here and I don't know how to factor things out and bring the 81 forth to get the expression in the form: "

**81(**to show that it's divisible by 81 for all n. Any help is appreciated.

*something here*)"P.S.

I might be able to do it this way:

10 × (81m+9k+10-k) = 810m+90k+100-10k

=> 81(10m+(90/81)k+(100/81)-(10/81)k)

But I don't feel that this is the right way...

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