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

In summary, "A positive integer divisible by 2019 the sum of whose decimal digits is 2019" refers to a number that can be evenly divided by 2019 and has a sum of 2019 when its digits are added together. There is a formula to find these numbers, which falls under the field of mathematics and specifically number theory. Real-world applications include cryptography. There is no limit to the size of these numbers, but it becomes increasingly difficult to find larger ones. There is also no known smallest number that meets this criteria.
  • #1
lfdahl
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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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  • #2
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 $$2\;\overbrace{0001\;0001\;\ldots\;0001}^{167\text{ blocks}}.$$

[/sp]
 
  • #3
Thankyou, Opalg, for your participation and - as always - for a clever answer! (Yes)
 

1. What is the smallest positive integer that is divisible by 2019 and has a sum of decimal digits equal to 2019?

The smallest positive integer that meets these criteria is 2019.

2. Is there only one integer that satisfies the conditions of being divisible by 2019 and having a sum of decimal digits equal to 2019?

No, there are infinitely many integers that meet these conditions. Some examples include 4038, 6057, and 8076.

3. Can you explain why the sum of decimal digits must be 2019 for an integer to be divisible by 2019?

This is because the divisibility rule for 2019 states that the sum of the first three digits must be divisible by 2019. Since 2019 is a four-digit number, the sum of its digits must also be divisible by 2019.

4. Is there a pattern or formula for finding integers that are divisible by 2019 and have a sum of decimal digits equal to 2019?

Yes, there is a pattern. The integers that meet these criteria can be written as 2019n, where n is any positive integer. For example, 2019, 4038, 6057, etc. are all divisible by 2019 and have a sum of decimal digits equal to 2019.

5. Can an integer be divisible by 2019 and have a sum of decimal digits equal to a number other than 2019?

Yes, it is possible. For example, the integer 20190 is divisible by 2019 and has a sum of decimal digits equal to 12. However, this is not as common as having a sum of decimal digits equal to the number itself (2019).

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