Base

Number Theory Level 2

If the decimal number 212 is written in base $n$ , then it is in form of $21212 _n$ , find $n$ .

3 4 7 2 9

1 solution

Alex G
May 27, 2016

Since the number uses 2 as a digit, $n>2$ .

The number is equivalent to $2+n+2n^2+n^3+2n^4$ . As such, $n^4<212$ . Note that $4^4=256$ , therefore $n<4$ . Since $2<n<4$ and n is an integer, $\boxed {n=3}$ .

Never thought of it like this. Good Job.

Hana Wehbi - 5 years ago

Thank you again. Bases are a passion of mine.

Alex G - 5 years ago

You are welcome. I will try to post more nice problems about bases.

Hana Wehbi - 5 years ago

Base

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