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ID Number And Digit Arrangement (Posted on 2008-08-19) Difficulty: 2 of 5
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.

See The Solution Submitted by K Sengupta    
Rating: 4.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Hmmmmm.... (spoilers) | Comment 1 of 9

I might be missing something but I get four possible ID numbers:

4  1  3  0  9  5  2  8  6  7
4  1  6  0  3  5  7  2  8  9
1  4  3  0  9  5  2  8  6  7
1  4  6  0  3  5  7  2  8  9

and don't know a way of eliminating two.

Yes, I used a program:

CLS

FOR b = 0 TO 9
 used(b) = 1

FOR c = 0 TO 9
 IF used(c) = 0 THEN
   used(c) = 1

FOR d = 0 TO 8 STEP 2
 IF used(d) = 0 AND c > d THEN
   used(d) = 1

FOR e = 0 TO 9
 IF used(e) = 0 THEN
  IF (c + d + e) MOD 3 = 0 THEN
   used(e) = 1

FOR f = 0 TO 5 STEP 5
 IF used(f) = 0 THEN
   used(f) = 1

FOR g = 0 TO 9
 IF used(g) = 0 THEN
  IF (100 * e + 10 * f + g) MOD 7 = 0 THEN
   used(g) = 1

FOR h = 0 TO 9
 IF used(h) = 0 THEN
  IF (100 * f + 10 * g + h) MOD 11 = 0 THEN
   used(h) = 1


FOR i = 0 TO 9
 IF used(i) = 0 THEN
  IF (100 * g + 10 * h + i) MOD 13 = 0 THEN
   used(i) = 1

FOR j = 0 TO 9
 IF used(j) = 0 THEN
  IF (100 * h + 10 * i + j) MOD 17 = 0 THEN
   used(j) = 1

   FOR a = 0 TO 9
     IF used(a) = 0 THEN EXIT FOR
   NEXT a
   PRINT a; b; c; d; e; f; g; h; i; j

   used(j) = 0
  END IF
 END IF
NEXT j

   used(i) = 0
  END IF
 END IF
NEXT i
   used(h) = 0
  END IF
 END IF
NEXT h

   used(g) = 0
  END IF
 END IF
NEXT g

   used(f) = 0
 END IF
NEXT f

   used(e) = 0
  END IF
 END IF
NEXT e

   used(d) = 0
 END IF
NEXT d
   used(c) = 0
 END IF
NEXT c

 used(b) = 0
NEXT b

 


  Posted by Charlie on 2008-08-19 12:39:09
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