For what values of $x$ is the number $121$ a square in base $x$?
This is a very easy question, but I still think it's pretty fun. Perhaps it's something to pass on to younger people who are taking algebra.
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Sign up to join this communityFor what values of $x$ is the number $121$ a square in base $x$?
This is a very easy question, but I still think it's pretty fun. Perhaps it's something to pass on to younger people who are taking algebra.
All of them (except base $2$ which can't have $121$.)
Added pedantry:
If you're ok with negative bases, the answer is all positive and negative bases except $-2$, $-1$, and $2$. Base $0$ and $1$ make no sense at all to me (and $-1$ very little) so I would also claim it doesn't hold for these, mostly because "holding" isn't really provably true or false, but also because 121 is not a valid number in those bases, as also it is not valid in base 2 and -2.
Also, I am considering integer bases only.
Why is it a square for these bases?
For whatever base $n$ you're working in, simply expanding the number $121$, it means $n^2 + 2n + 1$. And that is $(n+1)^2$ as a matter of algebra. And that is a square.
n^2 - 2n + 1
which reduces to (n-1)^2
so the value will still be a square if x < -2. Therefore, the complete answer is Integers <-2 and > 2
$\endgroup$
– Engineer Toast
Apr 22 '15 at 15:15
The base-$x$ number $121$ is a square
whenever $x\geq 3$.
For $x\geq 3$, we can multiply in base $x$: \begin{array}{ccc}&1&1\\\times&1&1\\\hline&1&1\\1&1&\\\hline1&2&1\end{array}We never needed to "carry", so this computation looks the same for every $x$.