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Round Results (Posted on 2013-08-05) Difficulty: 2 of 5
In how many ways can you select a set of six distinct integers, each having absolute value below 51 to yield a product which is an integer power of 10?

No Solution Yet Submitted by Ady TZIDON    
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Solution computer solution | Comment 1 of 4

DEFDBL A-Z
CLS
FOR a = 1 TO 45
IF a MOD 3 > 0 AND a MOD 7 > 0 AND a MOD 11 > 0 AND a MOD 13 > 0 THEN
FOR b = a + 1 TO 46
IF b MOD 3 > 0 AND b MOD 7 > 0 AND b MOD 11 > 0 AND b MOD 13 > 0 THEN
FOR c = b + 1 TO 47
IF c MOD 3 > 0 AND c MOD 7 > 0 AND c MOD 11 > 0 AND c MOD 13 > 0 THEN
  abc = a * b * c
FOR d = c + 1 TO 48
IF d MOD 3 > 0 AND d MOD 7 > 0 AND d MOD 11 > 0 AND d MOD 13 > 0 THEN
FOR e = d + 1 TO 49
IF e MOD 3 > 0 AND e MOD 7 > 0 AND e MOD 11 > 0 AND e MOD 13 > 0 THEN
FOR f = e + 1 TO 50
  IF f MOD 3 > 0 AND f MOD 7 > 0 AND f MOD 11 > 0 AND f MOD 13 > 0 THEN
  prod = abc * d * e * f
  s$ = LTRIM$(STR$(prod))
  IF LEFT$(s$, 1) = "1" THEN
    good = 1
    FOR i = 2 TO LEN(s$)
       IF MID$(s$, i, 1) <> "0" THEN good = 0: EXIT FOR
    NEXT i
    IF good THEN ct = ct + 1: PRINT a; b; c; d; e; f, prod
  END IF
END IF
NEXT f
END IF
NEXT e
END IF
NEXT d
END IF
NEXT c
END IF
NEXT b
END IF
NEXT a

PRINT ct

finds these 18 positive results:

 1  2  4  5  10  25          10000
 1  2  4  10  25  50         100000
 1  2  5  8  25  50          100000
 1  2  5  10  20  50         100000
 1  2  5  10  25  40         100000
 1  2  10  25  40  50        1000000
 1  4  5  10  20  25         100000
 1  4  5  25  40  50         1000000
 1  4  10  20  25  50        1000000
 1  5  8  20  25  50         1000000
 1  5  10  16  25  50        1000000
 1  5  10  20  25  40        1000000
 1  10  20  25  40  50       10000000
 2  4  5  20  25  50         1000000
 2  5  8  10  25  50         1000000
 2  5  20  25  40  50        10000000
 4  5  10  25  40  50        10000000
 5  8  10  20  25  50        10000000


 
Since the integers need not be positive, but the product, being a power of positive 10 does need to be positive, any 2 or 4 or all six of the integers could be negative.

So each of the 18 shown results can be turned into a set of positive and negative numbers:

All positive:      1
C(6,2):           15
C(6,4):           15
All negative:      1
                 ---
                  32
                 


The final answer is 18*32 = 576.             
                 

Edited on August 5, 2013, 9:54 pm
  Posted by Charlie on 2013-08-05 13:36:27

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