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Prime divisors (Posted on 2005-01-04) Difficulty: 3 of 5
6300846559 is such that 6 is divisible by 2; 63, by 3; 630, by 5; 6300, by 7; and, in general, if you take its first N digits, it will be divisible by the N-th prime.

There is only one other such 10 digit number: can you find it?

See The Solution Submitted by e.g.    
Rating: 3.7500 (4 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution re: Solution (VB program) | Comment 2 of 11 |
(In reply to Solution (VB program) by Penny)

Rather than have code to test each level separately, the following QuickBasic code uses recursion to add a level at a time, calling the same code over again:

DECLARE SUB buildOn (s$)
CLEAR , , 5000
DATA 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61
DEFDBL A-Z
DIM SHARED prime(18)

FOR i = 1 TO 18
  READ prime(i)
NEXT

buildOn ""

SUB buildOn (s$)
  l = LEN(s$) + 1
  had = 0
  FOR i = 0 TO 9
    d$ = LTRIM$(STR$(i))
    s1$ = s$ + d$
    v = VAL(s1$)
    q = INT(v / prime(l))
    r = v - q * prime(l)
    IF r = 0 THEN
     had = 1
     IF l = UBOUND(prime) THEN
      PRINT s1$; "*"
     ELSE
      buildOn s1$
     END IF
    END IF
  NEXT i
  IF had = 0 THEN
   PRINT USING "## &"; LEN(s$); s$
  END IF
END SUB

This version shows all the possible numbers, (up to 18 digits, but none need that), as they are built up.  Below, I've left out the ones with leading zeros; none of those reached 10 digits, except 000000... which could go on forever. Each string is preceded by its length; indeed in two cases it reaches 10 and 10 is never exceeded.

 4 2100
 9 210769470
 8 21560592
 7 2401369
 4 2408
 7 2450890
 5 24574
 6 270270
 6 270933
 7 2751195
 8 27588627
 8 42009551
 7 4207585
 4 4256
 6 450125
 7 4508927
 8 45507812
 5 45573
 5 48026
 6 480922
 7 4851086
 7 4858753
 6 600600
 7 6055536
10 6300846559
 7 6307476
 5 63569
 6 660114
 7 6608818
 9 665067264
 5 66572
 5 69025
 6 690911
 4 6951
 7 6958644
 4 8106
 8 81554270
 8 84007379
 8 84073670
 6 845689
 6 870103
 7 8708709
 7 8750563
10 8757193191

  Posted by Charlie on 2005-01-04 16:00:11
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