A positive integer divisible by 2019 with a decimal digit sum of 2019 exists. The construction involves the number 4038 followed by 167 blocks of 2019, yielding a total digit sum of 2019. The integer is confirmed as a multiple of 2019, with the quotient being 2 followed by 167 blocks of 0001. This solution effectively demonstrates the required properties through a systematic approach.
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[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 $$2\;\overbrace{0001\;0001\;\ldots\;0001}^{167\text{ blocks}}.$$lfdahl said:Prove the existence of a positive integer divisible by $2019$ the sum of whose decimal digits is $2019$.Source: Nordic Math. Contest