Digit addition

Number Theory Level 3

Let $s(n)$ be the digit sum of $n$ . For example, $s(2019)=2+0+1+9=12$ .

Also, define a digit-addition series :

$a_{k,n} = \begin{cases} n & \text{for } k=1 \\ a_{k-1,n}+s(a_{k-1,n}) & \text{otherwise} \end{cases}$

Do there exist positive integers $i$ and $j$ such that $a_{i,819} = a_{j,255}$ ?

No Yes

1 solution

Henry U
Mar 10, 2019

$9 | 819=a_{1,819} \Rightarrow 9 | s(819) \Rightarrow 9 | 819+s(819)=a_{2,819}$

The same argument proves (by induction) that any number $a_{i,819}$ will be divisible by 9.

Similarly, we can prove that any number $a_{j,255}$ will be divisible by 3, but not by 9. This is because $255 = 3 \cdot 5 \cdot 17$ .

So, all numbers $a_{i,819}$ are divisible by 9, but no number $a_{j,255}$ is, so they can never be equal.