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Anu refuses to tell anyone if she is working in a 1←10 system, or a 1←5 system, or any other system.  She makes everyone call it a 1 ← x system but won’t tell anyone what x stands for.

We know that boxes in a 1←10 have values that are powers of ten: 1, 10, 100, 1000, 10000…

And boxes in a 1←5 system are powers of five: 1, 5, 25, 125, 625…

So Anu’s system, whatever it is, must be powers of x: $1, x, x^2, x^3, x^4 \dots$

When Anu writes $2556 _x$ she must mean:

\[2x^3 + 5x^2 + 5x + 6.\]

And when she writes $12_x$ she means:

\[x + 2.\]

Anu decides to compute $2556_x \div 12_x$.

She obtains:

\[ (2x^3+5x^2+5x+6) \div (x+2) = 2x^2+x+3.\]

Anu later tells use that she really was thinking of a 1←10 system so that x does equal ten. Then her number 2556 really was two thousand, five hundred and fifty six and 12 really was twelve. Her statement: \[(2x^3+5x^2+5x+6)\div(x+2)=2x^2+x+3\] is actually 2556 ÷ 12 = 213.

Uh Oh! Anu has changed her mind. She now says she was thinking of a 1←11 system.

Now $2556_x$ means $2\cdot 11^3+5\cdot 11^2+5\cdot 11 +6 = 3328_{\text{ten}}$.

Similarly, $12_x$ means $1\cdot 11+2 =13_{\text{ten}}$, and $213_x$ means $2\cdot11^2+1\cdot 11+3 = 256_{\text{ten}}$.

So Anu’s computation $2556_x\div 12_x = 213_x$ is actually the (base 10) statement:

3328 ÷ 13 = 256.