A positive integer divisible by 2019 the sum of whose decimal digits is 2019.

[lfdahl]

Prove the existence of a positive integer divisible by $2019$ the sum of whose decimal digits is $2019$.Source: Nordic Math. Contest

 

[Opalg]

Prove the existence of a positive integer divisible by $2019$ the sum of whose decimal digits is $2019$.Source: Nordic Math. Contest

[sp]$2019$ has digital sum $12$. Twice $2019$ is $4038$, which has digital sum $15$. Also, $$2019 = 15 + 2004 = 15 + 12\cdot167.$$ So the number $$4038\;\overbrace{2019\;2019\;\ldots\;2019}^{167\text{ blocks}},$$ whose decimal expansion consists of $4038$ followed by $167$ blocks of $2019$, has decimal sum $2019$. It is clearly a multiple of $2019$, the quotient being $$2\;\overbrace{0001\;0001\;\ldots\;0001}^{167\text{ blocks}}.$$

[/sp]

 

[lfdahl]

Thankyou, Opalg, for your participation and - as always - for a clever answer! (Yes)