Determine the total number of quadruplets (A,B, C, D) of positive integers with A ≤ B ≤ C ≤ D ≤ 25, such that (A+B)*(C+D) is divisible by |A*D – B*C|, whenever A*D ≠ B*C.

Note: |x| refers to the absolute value of x.

DEFDBL A-Z
FOR a = 1 TO 25
FOR b = a TO 25
FOR c = b TO 25
FOR d = c TO 25
  num = (a + b) * (c + d)
  den = ABS(a * d - b * c)
  IF a * d = b * c THEN
    PRINT TAB(40); a; b; c; d
    ct0 = ct0 + 1
  ELSE
    IF num MOD den = 0 THEN
     PRINT a; b; c; d
     ct = ct + 1
    END IF
  END IF
NEXT
NEXT
NEXT
NEXT
PRINT ct, ct0

finds that 4271 sets satisfy all the criteria, while 462 sets have A*D = B*C.  It could be said that all 4733 sets are such that whenever A*D  not= B*C, (A+B)*(C+D) is divisible by |A*D - B*C|.

Edited on February 6, 2011, 5:37 pm

Edited on February 6, 2011, 5:39 pm

Posted by Charlie on 2011-02-06 17:36:56