Each of the ID numbers issued to Mr.Cooper and Mr. Duncan is of the form ABCDEFGHIJ, with each of the letters representing a different digit from 0 to 9 inclusively, such that:

(i) BCD is divisible by 2.

(ii) CDE is divisible by 3.

(iii) DEF is divisible by 5.

(iv) EFG is divisible by 7.

(v) FGH is divisible by 11.

(vi) GHI is divisible by 13.

(vii) HIJ is divisible by 17.

Determine the ID numbers issued to each of the gentlemen, given that the ID number of Mr. Cooper is greater than that of Mr. Duncan.

Note: A is not 0, and C is greater than D.

*** While a solution may be trivial with the aid of a computer program, show how to derive it without one.

I started with BCD, EFG, and HIJ: calculating all combinations of nine unique digits with BCD divisible by 2, EFG by 7, and HIJ by 17.  Then, using those assignments, I tested GHI for divisibility by 13, FGH by 11, DEF by 5 and CDE by 3 (using these sequences to reduce calculations.

There were six combinations: for each the unused digit was assigned to A.  For two of these (1 406 357 289 and 4 106 357 289) C was greater than D, so excluded.  This left four combinations, rather than the two implied by the problem text.

A.BCD.EFG.HIJ

1.460.357.289

1.430.952.867

4.160.357.289

4.130.952.867

I'll post these, then see what previous commentaries did with the extras, or what I missed in the specs.

Posted by ed bottemiller on 2008-08-20 11:33:24