Expert: Scotto - 11/11/2008

Question
QUESTION: Scott, I typed in the question poorly. Here it is again.

Prove that 81 divides (10^(n+1) -9n -10) for every non negative integer n.

The first time I didn't show that it was 10 to the n+1 power, that is probably why it wouldn't work.

Thank you.

ANSWER: N    Answer   Result
1   81   1
2   972   12
3   9963   123
4   99954   1234
5   999945   12345
6   9999936   123456

Here you can see that I tried 1,2,3,4,5 and 6, in order, and these were the results.

That makes me think we need see how to get from one term to the next.

(A) If we take 10^(n+2) - 9(n+1) - 10 - (10^n+1) - 9n - 10) we get
9*10^(n+1) - 9.  This can be seen to be 81, 891, 8991, ...

(B) We know that a string of n 1's * 81 is as string on n 8's followed by a zero + a string of n 1's.  Add these together and the starting digit is an 8 and the ending digit is a 1 with all 9's in the middle.

Since the difference in (A) is the same as the product in (B), it is true for all n.

---------- FOLLOW-UP ----------

QUESTION: I think that I understand the a part, but don't quite follow what you did in the b part. What do you mean by a string of n 1's and a string of n 8's.

Answer
It's a pity I can't line up numbers.  The numbers in this letter are lined up in columns on my end.

(A) Term n+1 is 10^(n+2) - 9(n+1) - 10.
Term n is (10^n+1) - 9n - 10).
The difference between them is
10^(n+2) - 9(n+1) - 10 - (10^n+1) - 9n + 10 (I think I acceidentally put a -10 for the last term note that minus a minus is a plus for the last 10).

The result includes 10^(n+1) - 10^n = 9*10^(n+1),
9(n+1) - 9n = 9, and -10 + 10 = 0.  Overall this is 9*10^(n+1) - 9.

This can be seen to be
N
1 90-9=81
2 900-9=891
3 9000-9=8991
...
That's what I mean when I say it starts with 8 and ends with 1.

(B) We know that a string of n 1's (ex 11, 111, 1111) * 81 is a string of 9's between a 8 at the start and a 1 at the end (ex 891; 8,991; 89,991).

Since the difference in (A) is the same as the product in (B), it is true for all n.

I inspected this answer better and hopefully it is understandable now.