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I was recently asked why the following results holds

\begin{align} 1 \times 8 + 1 &= 9 \\ 12 \times 8 + 2 & = 98 \\ 123 \times 8 + 3 & = 987 \\ 1234 \times 8 + 4 & = 9876 \\ 12345 \times 8 + 5 & = 98765 \\ 123456 \times 8 + 6 & = 987654 \\ 1234567 \times 8 + 7 & = 9876543 \\ 12345678 \times 8 + 8 & = 98765432 \\ 123456789 \times 8 + 9 & = 987654321. \\ \end{align}

I honestly had some trouble giving my friend an answer. After filling a writing block with pencil equations, I found a hint in a number theory book for a similar result and used the fact that any number can be represented as $N = a_n10^n+a_{n-1}10^{n-1}+\ldots a_110 + a_0$ and playing around with the above equations, one can see algebraically why this holds.

From my notes, I see that I showed that

$$(10-2)(1 \cdot 10^{n-1} + 2 \cdot 10^{n-2} + \ldots + n \cdot 10^0) + n = $$ $$(10-1)10^{n-1} + (10-2)10^{n-2} + \ldots (10-n)10^0.$$.

This is not very satisfactory. I know from number theory, that there is a result $10 \equiv 1 \pmod9$ and I have seen some interesting consequences of it. Can this result be used to show why the 'pyramid' holds? Something similar perhaps?

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