Substitute each of the capital letters in bold by a different base ten digit from 0 to 9 to satisfy this set of alphametic relationships. None of the numbers can contain any leading zero.

• NATE is divisible by 3, and:
• ATEN is divisible by 7, and:
• ANTE is divisible by 11, and:
• NEAT is divisible by 17.

What are the possible value(s) that NEAT can assume?

The two sets of values followed by lines with their values mod 3, mod 7, mod 11 and mod 17:

 9270  2709  2970  9027
 0  0  0  0
 2  0  2  4
 8  3  0  7
 5  6  12  0

 9417  4179  4917  9741
 0  0  0  0
 2  0  3  4
 1  10  0  6
 16  14  4  0

FOR a = 1 TO 9
 used(a) = 1
FOR n = 1 TO 9
 IF used(n) = 0 THEN
   used(n) = 1
FOR t = 0 TO 9
 IF used(t) = 0 THEN
   used(t) = 1
FOR e = 0 TO 9
 IF used(e) = 0 THEN
   used(e) = 1

nate = n * 1000 + a * 100 + t * 10 + e
neat = n * 1000 + e * 100 + a * 10 + t
ante = a * 1000 + n * 100 + t * 10 + e
aten = a * 1000 + t * 100 + e * 10 + n
IF nate MOD 3 = 0 THEN
  IF aten MOD 7 = 0 THEN
  IF ante MOD 11 = 0 THEN
  IF neat MOD 17 = 0 THEN
   PRINT nate; aten; ante; neat
   PRINT nate MOD 3; aten MOD 3; ante MOD 3; neat MOD 3
   PRINT nate MOD 7; aten MOD 7; ante MOD 7; neat MOD 7
   PRINT nate MOD 11; aten MOD 11; ante MOD 11; neat MOD 11
   PRINT nate MOD 17; aten MOD 17; ante MOD 17; neat MOD 17
   PRINT
  END IF
  END IF
  END IF
END IF

   used(e) = 0
 END IF
NEXT
   used(t) = 0
 END IF
NEXT
   used(n) = 0
 END IF
NEXT
 used(a) = 0
NEXT

 

Posted by Charlie on 2009-09-27 13:30:42