What pairs of the digits 123456789 work as last digits?
(1,3) (2,6) (3,9) (4,2) (6,8) (7,1) (8,4) (9,7)
(4,2), xxx4 / xxx(x+1)2, leaves 1356789, possible (5,6) (6,9) (8,5) (9,8)
(4,2), (5,6), xx54 / xxx62, leaves 13789, possible (9,8)
(4,2), (5,6), (9,8), x954 / xx862, leaves 137, no solution
(4,2), (6,9), xx64 / xx(x+1)92, leaves 13578, possible (8,5)
(4,2), (6,9), (8,5), x864 / x(x+2)592, leaves 137, no solution
(4,2), (8,5), xx84 / xx(x+2)52, leaves 13679, possible (3,1), (7,3)
(4,2), (8,5), (3,1), x384, /x(x+1)152, leaves 679, no solution
(4,2), (8,5), (7,3), x784 / x(x+2)352, leaves 169, no solution
(4,2), (9,8), xx94 / xx(x+2)82, leaves 13567, possible (1,5), (3,1), (5,7)
(4,2), (9,8), (1,5), x194 / xx582, leaves 367, no solution
(4,2), (9,8), (3,1), x394 / x(x+1)182, leaves 567, no solution
(4,2), (9,8), (5,7), x594 / x(x+1)782, leaves 136, no solution
(1,3), xxx1 / xxxx3, leaves 2456789, possible (2,6), (4,2), (6,8), (8,4), (9,7)
(1,3), (2,6), xx21 / xxx63, leaves 45789, possible (8,4), (9,7)
(1,3), (2,6), (8,4), x821 / x(x+2)463, leaves 579, no solution
(1,3), (2,6), (9,7), x921 / x(x+2)763, leaves 458, no solution
(1,3), (4,2), xx41 / xxx23, leaves 56789, no solution
(1,3), (6,8), xx61 / xx(x+1)83, leaves 24579, possible (2,7), (7,2)
(1,3), (6,8), (2,7), x261 / xx783, leaves 459, no solution
(1,3), (6,8), (7,2), x761 / x(x+2)283, leaves 459, no solution
(1,3), (8,4), xx81 / xx(x+2)43, leaves 25679, possible (5,7)
(1,3), (8,4), (5,7), x581 / x(x+1)743, leaves 269, no solution
(1,3), (9,7), xx91 / xx(x+2)73, leaves 24568, possible (2,8), (8,6)
(1,3), (9,7), (2,8), x291 / xx873, leaves 456, no solution
(1,3), (9,7), (8,6), x891 / x(x+2)673, leaves 245, no solution
(2,6), xxx2 / xxxx6, leaves 1345789, possible (1,3), (3,9), (7,1), (8,4), (9,7)
(2,6), (1,3), xx12 / xxx36, leaves 45789, possible (8,4), (9,7)
(2,6), (1,3), (8,4), x812 / x(x+2)436, leaves 579, no solution
(2,6), (1,3), (9,7), x912 / x(x+2)736, leaves 458, no solution
(2,6), (3,9), xx32 / xxx96, leaves 14578, possible (7,1), (8,4)
(2,6), (3,9), (7,1), x732 / x(x+2)196, leaves 458, no solution
(2,6), (3,9), (8,4), x832 / x(x+2)496, leaves 157, possible (5,7)
(2,6), (3,9), (8,4), (5,7), 5832 / 17496
Check. Does 5832 / 17496 = 1/3? Yes.
Answer: 5832 / 17496
Other solutions: I will leave it as an exercise to the reader to look for any additional solutions.