Which $N$ are multiples of $2020$ with all distinct digits where swapping any two of them makes $N$ not a multiple of $2020$?

Question

The question

I have attempted problem 3 from the Senior O-level Tournament of Towns paper, Fall 2020. The question is as follows:

A positive integer number N is divisible by 2020. All its digits are different and if any two of them are swapped, the resulting number is not divisible by 2020. How many digits can such a number N have?

Sergey Tokarev

My attempt

Let $n$ be the number of digits of $N$. I initially noticed that $4 \leq n \leq 10$, since for all digits of $N$ to be distinct we can have at most $10$ digits. This gave me the impression that this problem could be solved by investigating each of these cases.

Let us first write $N$ as $2020k = 2000k + 20k$. Let $2k$ have digits $d_1d_2\cdots d_m$, then $N = (d_1d_2\cdots d_m)000 + (d_1d_2\cdots d_m)0$.

Case n = 4: Then $2k = d_1$ and $N = d_1000 + d_10 = d_10d_10$, so $N$ must have a repeating digit.

Case n = 5: Then $2k = d_1d_2$ and $N =d_1d_1000 + d_1d_20 = d_1d_2d_1d_20$, so $N$ must have a repeating digit.

Case n = 6: Then $2k = d_1d_2d_3$ and $N = d_1d_2d_3000 + d_1d_2d_30 = d_1d_2(d_3 + d_1)d_2d_30$, so $N$ must have a repeating digit.

This argument no longer works for $n \geq 7$. But we do know that

1. $d_m \in \{0, 2, 4, 6, 8\}$ since $2k = d_1d_2\cdots d_m$, and
2. $N$ must end in $0$.

I used this to find numbers for $n = 7, 8, 9$ and $10$ with all distinct digits that are also divisible by $2020$ because they satisfy the observations listed above.

\begin{array}{|c|c|c|c|} \hline n & 7 & 8 & 9 & 10\ \\ \hline N & 4981320 & 64975320 & 869753420 & 2537618940 \\ \hline \end{array}

Claim: If $N$ is a multiple of $2020$ and has all distinct digits, then swapping any two digits results in a number $N'$ where $2020 \not \mid N'$.

proof. Let $N = a_na_{n-1}\cdots a_j\cdots a_i\cdots a_1$ and $N' = a_na_{n-1}\cdots a_i\cdots a_j\cdots a_1$ for any $i < j$. If $2020 \mid (N' - N)$ then $2020 \mid N'$. Suppose, WLOG, that $a_i > a_j$ so that $N' > N$, then $$N' - N = 10^{j - 1}(a_i - a_j) - 10^{i - 1}(a_i - a_j) = (a_i - a_j)(10^{j - 1} - 10^{i - 1})$$

So $N' - N$ must be a multiple of $10$. Therefore, for the claim to hold, $N'$ must not be a multiple of $4$ or $101$, since $2020 = 2^2 \times 5 \times 101$. We know that $1 \leq a_i - a_j \leq 9$, thus we can safely conclude that $N' - N$ is never a multiple of $101$. Therefore, $2020 \not \mid (N' - N)$.

Therefore, $N$ can have $7, 8, 9$ or $10$ digits and we are done.

I think my solution is correct, but I don't like that I had to manually find $N$'s for $7 \leq n \leq 10$. Is there a different way to argue for those cases or a more systematic way to find $N$'s for those cases? Or if you have a different way to solve the problem, I would also like to see it.

Answer 1

The error in your logic is that $N' - N = (a_i - a_j) (10^{i-1} ) ( 10^{j-i } - 1 ) $ could be a multiple of 101 when $ j-i = 4, 8$.

This yields a multiple of 2020 if $(a_i - a_j) (10^{i-1} )$ is a multiple of 20.

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If the number has 7 or more digits, then swapping the 1st and 5th digit yields a difference of $(a_n - a_{n-4}) \times 9999 \times 10^{n-5} $, which is a multiple of 2020.

Hence there are no solutions with 7 or more digits.

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Your arguments for $n = 4, 5$ are correct.
However, it fails for $n=6$ in the event that $ d_3 + d_1 \geq 10$, as the carry over makes the second digit $d_2 + 1$, so it need not be a repeated digit with the other $d_2$. (It could still repeat with another digit though).
As an explicit example, $ 2020 \times 377 = 761540$ has distinct digits. I found it through trial and error.

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If remains to find an example for $n=6$. If no such example can be found, then the answer to the question is "no digits" (which seems unlikely as an answer).

Suppose $N = a_6a_5a_4a_3a_2a_1$. Then, to ensure that the swap doesn't yield a difference that is a multiple of 2020,

• In order to be a multiple of 101, we just need to concern ourselves with terms that are 4 apart.
• We then need to check that $(a_6 - a_2)10$ and $(a_5 - a_1)$ are not multiples of 20.

These are satisfied by $761540$, so it is a valid example.

Hence the number must have 6 digits.

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To find the counterexample without excessive trial and error, since (using OP's notation) $\overline{d_1d_2d_3} \times 1010 = \overline{d_1d_2(d_3+d_1)d_2d_30}$, then the conditions that we need are (I might have missed some):

• $0, d_1, d_2, d_3 $ are distinct digits
• $d_3$ is a multiple of 2
• $d_1 - d_3$ is not a multiple of 2.
• $d_2$ is not a multiple of 20.
• $d_1 + d_3 \geq 11$ (cannot be 10 which leaves the digit of 0)
• $d_2 + 1 \neq d_1, d_3, 10$
• $d_1 + d_3 - 10 \neq d_2 $

These are satisfied by $ 754 = 377 \times 2$, which is why the example works. Smaller values include $348$, which yields $351480$.