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?

## Comments

Sort by:

TopNewestI'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 · 1 year, 1 month ago

Log in to reply

– Raja Metronetizen · 1 year, 1 month ago

I think, both of them are same.Log in to reply

– Ameya Daigavane · 1 year, 1 month ago

So this is what you meant, right?Log in to reply