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 Gross Division (Posted on 2006-02-20)
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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 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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