Once they learn how to use a computer, it's amazing how many people make it do their thinking for them. This is (almost) just a simple logic problem.
Although you can could use the computer to generate the possible solutions and eliminate duplicate numbers, I found it just as easy to do it with my text editor.
o Since 0 cannot be used, #5 must be a 5.
o #1 can be 1, 3, 7 or 9.
o #2 can be 2, 4, 6 or 8.
o #3 can be 3, 5, 7 or 9.
o #1-3 must be a multiple of 3. That gives 20 possibilities:
123 321 723 921
129 327 729 927
147 369 741 963
183 381 783 981
189 387 789 987
o Since #1-4 must be a multiple of 4, #3-4 must also be a multiple of
4, so #4 must be 2 or 6 (since #3 is odd).
o Since #4 is 2 or 6, and #5 is 5, there are only 6 possibilies for
the #4-6 range:
254 652
256 654
258 658
o Since #1-6 must be a multiple of 3 as well as 6, and since #1-3 are
already a multiple of 3, #4-6 must be a multiple of 3. This leaves
only three possibilities for #4-6:
258
654
658
o Since #1-8 must be a multiple of 8 and since #6 is either 4 or 8,
#7-8 must be a multiple of 8, so #7-8 can be:
16
32
72
96
o Since #4 and #8 are assigned 2 and 6, #2 and #6 must be 4 and 8.
o Since #2 must be 4 or 8, the possibilies for #1-3 are cut in half:
147 741
183 783
189 789
381 981
387 987
o Combine the possibilities for #1-3 and #4-6:
#1-3 #4-6
---- ----
147 258
183 654
189 658
381
387
741
783
789
981
987
147258 147654 (4) 147658
183258 (8) 183654 183658 (8)
189258 (8) 189654 189658 (8)
381258 (8) 381654 381658 (8)
387258 (8) 387654 387658 (8)
741258 741654 (4) 741658
783258 (8) 783654 783658 (8)
789258 (8) 789654 789658 (8)
981258 (8) 981654 981658 (8)
987258 (8) 987654 987658 (8)
o And eliminate repeated 4's and 8's:
147258 147658
183654
189654
381654
387654
741258 741658
783654
789654
981654
987654
o #8 must be 2 or 6 (whichever #4 isn't), and the possibilities for
#7-8 are limited to 16, 32, 72 and 96. So if #4 is a 2, #7-8 can be
16 or 96 and if #4 is a 6, #7-8 can be 32 or 72.
14725816 (1) 14725896
14765832 14765872 (7)
18365432 (3) 18365472
18965432 18965472
38165432 (3) 38165472
38765432 (3) 38765472 (7)
74125816 (1) 74125896
74165832 74165872 (7)
78365432 (3) 78365472 (7)
78965432 78965472 (7)
98165432 98165472
98765432 98765472 (7)
o And eliminate repeated digits:
14725896
14765832
18365472
18965432
18965472
38165472
74125896
74165832
78965432
98165432
98165472
98765432
o Using all nine digits once assures us that any number will be a
multiple of 9, so #9 is whatever is left over. The 12 possible
solutions fall neatly into place:
147258963
147658329
183654729
189654327
189654723
381654729
741258963
741658329
789654321
981654327
981654723
987654321
o These can be checked for divisibility by 7 with a calculator. :-)