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Gross Division (Posted on 2006-02-20) Difficulty: 3 of 5
Find the smallest positive integer n such that n has exactly 144 distinct positive divisors and there are 10 consecutive integers among them (Note: 1 and n are both divisors of n)

See The Solution Submitted by goFish    
Rating: 3.8000 (5 votes)

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Solution computer solution | Comment 10 of 14 |

DECLARE SUB factorIt ()
DEFDBL A-Z

CLEAR , , 5000

DIM SHARED n, done, rmndr
DIM SHARED fact(12), prime(6)
DIM SHARED nFact

DATA 2,3,5,7,11,13
FOR i = 1 TO 6: READ prime(i): NEXT

n = 144
done = 1: rmndr = n: nFact = 0

factorIt

SUB factorIt
  IF nFact = 0 THEN
    strt = 2
  ELSE
    strt = fact(nFact)
  END IF
  FOR i = strt TO SQR(rmndr)
   IF rmndr MOD i = 0 THEN
     rmndr = rmndr \ i
     done = done * i
     nFact = nFact + 1
     fact(nFact) = i
    
     factorIt
    
     rmndr = rmndr * i
     done = done \ i
     nFact = nFact - 1
   END IF
  NEXT
 
  done = done * rmndr
  nFact = nFact + 1
  fact(nFact) = rmndr
  rmndr = 1
 
  nbr = 1: good = 1
  FOR i = 1 TO nFact
   nbr = nbr * prime(nFact + 1 - i) ^ (fact(i) - 1)
   IF nbr > 100000000 THEN good = 0: EXIT FOR
  NEXT
  IF good THEN
    ct = 0
    FOR i = 1 TO nbr / 2
     r = nbr MOD i
     IF r = 0 THEN
      ct = ct + 1
      IF ct >= 10 THEN
       PRINT nbr, i; "****"
      END IF
     ELSE
      ct = 0
     END IF
    NEXT
  END IF
 
  rmndr = fact(nFact)
  nFact = nFact - 1
  done = done \ rmndr
END SUB

finds

110880        10 ****
110880        11 ****
110880        12 ****
138600        10 ****
138600        11 ****
138600        12 ****
645120        10 ****
241920        10 ****
272160        10 ****
201600        10 ****
151200        10 ****
264600        10 ****

Indeed the lowest number on the left is 110880.  The last factor in any series of factors greater than or equal to 10 is listed so this number is listed twice as it's divisible by 11 and 12 as well.


  Posted by Charlie on 2006-02-20 17:35:30
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