Matt Occhipinti, Mundia Mubyana, Melissa Mitchell
Correct solutions were also submitted by Mahendra Nambiar, Nathan Pauli. From outside Bradley University, correct solutions were submitted by high school student Ivica Nikolik, Will Troxel, Burkart Venzke, Monty Gray, Brian Laughlin, Baek Jongmin, Aaron David Kahn, Steve Prowse, Sorin Ionescu, Lorenzo Pozzoli, Emanuele Macri, Massimo Brignone, Ron Welch, Tim Kelley, Cyril Terakopiantz, Denis Borris, Philippe Fondanaiche, Allen Druze.
I intended the solution to use the digits 1 through 9, but neglected to say so. Using this restriction, the only solution is the integer 381654729. If you permit any nine digits, there are three other solutions, namely 381654720, 381654729, and 801654723. The more perverse solvers included the "solution" 081654327.
You can check each integer from 100,000,000 to 999,999,999, but it seems more efficient to use a few divisibility checks.
(a) d5 must be 0 or 5;The combination of these reduces the number of checks to a manageable number yielding the above solutions fairly quickly by hand.
(b) d2, d4, d6, d8 must be even;
(c) d1 + d2 + d3, d4 + d5 + d6, d7 + d8 + d9 must be divisible by 3;
(d) the two digit number d3d4 must be divisible by 4;
(e) the three digit number d6d7d8 must be divisible by 8.
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