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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(2): Similar solution | Comment 4 of 7 |
(In reply to re: Similar solution by Charlie)

Sometimes we need only the rightmost n-1 digits.  The modified algorithm:

    5   for N=2 to 32
   10     print N;" ";5^N;" ";:gosub 100:print Ans;" ";Ans//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
  152   V=val(right(K$,N-1)):R=V@(5^N)
  154   if R=0 then
  155     :Ans=V
  156   :else Ans=val(right(K$,N))
  180   return

finds

2   25   25   1
3   125   625   5
4   625   625   1
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   3759765625   77
12   244140625   275634765625   1129
13   1220703125   455322265625   373
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   6266367435455322265625   525661
24   59604644775390625   29479563236236572265625   494585
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 10:54:23
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