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Cubeless? (Posted on 2004-08-29) Difficulty: 3 of 5
Prove that for any positive integer n, there exists at least one multiple of 5^n that doesn't have any perfect cube digits (0, 1, or 8) in its decimal representation.

See The Solution Submitted by Federico Kereki    
Rating: 3.4000 (5 votes)

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re: Similar solution | Comment 3 of 7 |
(In reply to Similar solution by e.g.)

An implementation of that algorithm

    5   for N=2 to 32
   10     print N;" ";5^N;" ";:gosub 100:print val(right(K$,N))//5^N
   20   next
   30   end
  100   K=5^N:L=K:K$=cutspc(str(K))
  101   while len(K$)<N:K$="0"+K$:wend
  110   for I=1 to N
  120   if mid(K$,len(K$)-I+1,1)="0" or mid(K$,len(K$)-I+1,1) ="1" or mid(K$,len(K$)-I+1,1)="8" then
  130      :K=K+L:K$=cutspc(str(K))
  131      :while len(K$)<N:K$="0"+K$:wend
  140   L=L*10
  150   next
  160   print right(K$,N);" ";
  180   return

leads to the table below, which lists n, 5^n, the multiple of 5^n that contains no 0's, 1's or 8's, and what multiple that is of 5^n:

2   25  25  1
3   125  625  5
4   625  5625  9
5   3125  65625  21
6   15625  265625  17
7   78125  7265625  93
8   390625  97265625  249
9   1953125  697265625  357
10   9765625  6259765625  641
11   48828125  53759765625  1101
12   244140625  275634765625  1129
13   1220703125  5455322265625  4469
14   6103515625  76666259765625  12561
15   30517578125  463775634765625  15197
16   152587890625  7557525634765625  49529
17   762939453125  39432525634765625  51685
18   3814697265625  699222564697265625  183297
19   19073486328125  6593265533447265625  345677
20   95367431640625  47336673736572265625  496361
21   476837158203125  254633426666259765625  534005
22   2384185791015625  4364664554595947265625  1830673
23   11920928955078125  56266367435455322265625  4719965
24   59604644775390625  529479563236236572265625  8883193
25   298023223876953125  6569396555423736572265625  22043237
26   1490116119384765625  49764443933963775634765625  33396353
27   7450580596923828125  323633767664432525634765625  43437389
28   37252902984619140625  6553496457636356353759765625  175919081
29   186264514923095703125  64966366626322269439697265625  348785525
30   931322574615478515625  447736994363367557525634765625  480753937
31   4656612873077392578125  9662663773633539676666259765625  2075041245
32   23283064365386962890625  93647676543332636356353759765625  4022137081

  Posted by Charlie on 2004-08-30 10:42:26
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