Team selection tests for GMO 2018

Let $k$ be the number of digits of $N$ and let $N={\sum }_{i=0}^{k-1}10^i\cdot d_i$ where $d_i$ are digits. If $M$ is obtained by swapping the $i$-th and the $j$-th digits, then $$M-N=(10^j-10^i)\cdot (d_i-d_j).$$ Therefore $13\mid M-N$ implies that $d_i=d_j$ or $6\mid i-j$. Hence, a number $N$ satisfies the property given in the problem if and only if $13\mid N$, $d_i\ne d_j$ for all $i\ne j$ and no swap between $i$-th and $j$-th digits are possible if $6\mid i-j$.

- If $k\ge 8$, one can swap the 0th and the 6th digits, hence $N$ does not satisfy the property. - If $k=7$, one can swap the 0th and the 6th digits unless the 0th digit is 0, but if the 0th digit is 0, then no swap is possible.

Hence, $N$ satisfies the property if and only if $13\mid N$, all the digits are distinct and the 0th digit is 0. Therefore, $N\le 9876540$.

Considering all multiples of 13 that end with 0 below this level, one finds $N\in \{9876490,9876360,9876230,\dots \}$. Since the two greatest elements of this set do not have all distinct digits, the greatest integer $N=9876230$.