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  • +1 for a clear and concise mathematical explanation. And to answer the more general version of the question posed in the title, an example of a number not representable in base 10 is 1/3. Commented Apr 25, 2014 at 15:43
  • @Doval I do suspect there's a glitch in my reasoning or explanation that a more math oriented person could point out... but I do think I'm on the right track if that is the case. Commented Apr 25, 2014 at 15:49
  • "Relatively prime" in this case just means "not a factor of", right? Is there some deeper mathematical relationship I'm missing? Commented Apr 25, 2014 at 16:55
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    Ah, so as I understand it, n = 15 and b = 10 are not relatively prime ("share no common positive factors (divisors) except 1") because they share 5 as a factor. The key is that not all of the factors of 15 (5 and 3) are not also factors of 10. (Aside: is there a word to indicate numbers that do or don't share all common factors?) I think that is neatly wrapped up in your k, n, m equation, but to really wrap my head around it, I would need to see a 3d plot. Regardless, well deserved +1 to you. Commented Apr 25, 2014 at 17:08
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    @PatrickM: "Aside: is there a word to indicate numbers that do or don't share all common factors?": Any integer is a factor of itself, so if all factors of m are factors of n, then it trivially follows that m is a factor of n. One term for this, as you clearly know, is factor. Another is divisor. Commented Apr 26, 2014 at 20:10