# What is the radix (base) of number $(9)_r$ if its decimal representation is $(21)_{10}$?

I want to find $r$. The method that I followed is:

$(9\times r^0) = (2\times 10^1) + (1\times 10^0)$

which gives no solution.

• Based on this answer it sounds to me like "no solution" is correct, although I don't have the time right now to think about it more thoroughly. – tilper Jun 13 '17 at 13:57
• Are you sure the problem is stated correctly? Seems to me that $(9)_r$ is the number $9$ no matter what $r$ is, and that certainly isn't $21$. There is no radix that will make these values equal. You are right in saying that there is no solution to the problem as stated. – MPW Jun 13 '17 at 13:58
• Maybe it's meant that $(9)_{10} = (21)_4$, that would at least make sense. It would answer in what radix, the number 9 has the representation "21" – Henno Brandsma Jun 13 '17 at 14:02
• @MPW Yes, the problem is stated correctly. Actually my birthday cake could only hold 9 candles but I am 21, so I wanted to find a base for 9 that would justify this. – ahsan Jun 13 '17 at 14:08
• Also, the number of candles is just the number of candles (i.e. the cardinality of that set), it has no radix. A string representation does have a radix. So the idea is ill-posed. – Henno Brandsma Jun 13 '17 at 14:14