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

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A positive integer divisible by 2019 with a sum of decimal digits equal to 2019 can be constructed using the number 4038 followed by 167 blocks of 2019. The total digital sum of this number is confirmed to be 2019. The number is also a multiple of 2019, with the quotient being a sequence of 2 followed by 167 blocks of 1. This construction demonstrates the existence of such an integer. The solution showcases a clever approach to the problem.
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Prove the existence of a positive integer divisible by $2019$ the sum of whose decimal digits is $2019$.Source: Nordic Math. Contest
 
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lfdahl said:
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 $$2\;\overbrace{0001\;0001\;\ldots\;0001}^{167\text{ blocks}}.$$

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