MathDB

For each positive integer

k

we denote by

S(k)

the sum of its digits, for example

S(132)=6

and

S(1000)=1

. A positive integer

n

is said to be

<span class='latex-bold'>fascinating</span>

if it holds that

n = \frac{k}{S(k)}

for some positive integer

k

. For example, the number

11

<span class='latex-bold'>fascinating</span>

since

11 = \frac{198}{S(198)} (

since

\frac{198}{S(198)}=\frac{198}{1+9+8}=\frac{198}{18} = 11)

. Prove that there exists a positive integer less than

2021

and that it is not

<span class='latex-bold'>fascinating</span>

Problem Statement