Help : proof of a mathematical statement

How can I prove that any positive integer number or its next positive integer can be represented as the sum of an other positive integer and the sum of the digits of that other positive integer ?

I went forward with the steps that we assume a positive integer k and we can represent k as per below : $k= \sum_{n=0}^i {10}^{n} * {k}_{n}$ where $0 \leq k_1 , k_2 , ..., k_i \leq 9$

sum of the digits would be $k_1+k_2+....+k_i$ . what to do next?

#NumberTheory

Easy Math Editor

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Comments

I'm not sure I follow the question, are you saying,
Either $n$ or $n + 1$ can be represented as $m + Q(m)$ where $Q(m)$ is the sum of digits of $m$ ?

Ameya Daigavane - 5 years, 2 months ago

I think, both of them are same.

Sayan Chaudhuri - 5 years, 2 months ago

So this is what you meant, right?

Ameya Daigavane - 5 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$