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10 4 7 10 (Posted on 2015-11-03) Difficulty: 3 of 5
Consider decimal number 33. If converted to base 4 form it would be 201, which if regarded as 7 base number would yield 99 decimal.
Pay attention: 99=3*33.
219 is another number displaying the same feature (when converted to base 4 and interpreted as base 7 creates a multiple of the original decimal number).

Please list some additional samples.

No Solution Yet Submitted by Ady TZIDON    
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Some Thoughts A few | Comment 1 of 2
Any string of n digits written in base 7 instead of base 4 will be around (7/4)^(n-1) times bigger.

I wrote a little program dividing the string of digits in base 7 and dividing by base 4 an noting when this ratio was an integer.

Base 10 base 4 / 7 base 10 ratio
33 -> 201 -> 99 ratio = 3 
(7/4)^2 = 3.0625

219 -> 3123 -> 1095 ratio =5
(7/4)^3 = 5.359

840 -> 31020 -> 7560 ratio = 9
(7/4)^4 = 9.379

1055 -> 100133 -> 16880 ratio = 16
(7/4)^5 = 16.4

Searching up to 6 digits base 4 
interestingly there is one solution for each length (none for length 2 or 1)

  Posted by Jer on 2015-11-03 14:24:44
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