Problem #PRU-100225

Problems Number Theory Divisibility Divisibility of a number. General properties Divisibility rules

Problem

We call a $10$ -digit number interesting if it is divisible by $11111$ , and all its digits are different. How many interesting numbers does there exist?
Note that a number $k = a_{0} + 10 a_{1} + \dots + 10^{9} a_{9}$ is divisible by $11111$ if and only if a number $m = (a_{0} + a_{5}) + 10 (a_{1} + a_{6}) + \dots + 10^{4} (a_{4} + a_{9})$ is also divisible by $11111$ . This is because $100000 = 1 + 9 \times 11111$ and we subtract $99999 (a_{5} + 10 a_{6} + 100 a_{7} + 1000 a_{8} + 10000 a_{9})$ from the original number.

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