Old Competition in Slovenia No.1

[rsoldo]

Fraction $\frac{101010101}{110010011}$ is written in any base. Prove:

$$\frac{101010101}{110010011}=\frac{1010\overbrace{1\ldots1}^{n}0101}{1100\underbrace{1\ldots 1}_{n}0011}$$

and $n$ is odd number.

[TOTAL]

$$\frac{101010101}{110010011}=\frac{1010\overbrace{1\ldots1}^{n}0101}{1100\underbrace{1\ldots 1}_{n}0011}$$

Вопрос по условию: что такое стоит слева и что такое справа?

[Rak so dna]

rsoldo в сообщении #1578551 писал(а):

Fraction $\frac{101010101}{110010011}$ is written in any base. Prove:

$$\frac{101010101}{110010011}=\frac{1010\overbrace{1\ldots1}^{n}0101}{1100\underbrace{1\ldots 1}_{n}0011}$$

and $n$ is odd number.

$\frac{1+b^2+b^4+b^6+b^8}{1+b+b^4+b^7+b^8}=\frac{1+b^2+\frac{b^4(b^n-1)}{b-1}+b^{n+5}+b^{n+7}}{1+b+\frac{b^4(b^n-1)}{b-1}+b^{n+6}+b^{n+7}}$

[gris]

По-видимому речь идёт о равенствах

101010101 / 110010011 = 9091/9901
1010110101 / 1100110011 = 9091/9901
10101110101 / 11001110011 = 9091/9901
101011110101 / 110011110011 = 9091/9901
1010111110101 / 1100111110011 = 9091/9901
10101111110101 / 11001111110011 = 9091/9901
101011111110101 / 110011111110011 = 9091/9901
1010111111110101 / 1100111111110011 = 9091/9901
10101111111110101 / 11001111111110011 = 9091/9901

Но почему n только нечётное

[TOTAL]

Если в обеих частях (какое-то) одинаковое основание, то

$$\frac{1010\overbrace{1\ldots1}^{n}0101}{1100\underbrace{1\ldots 1}_{n}0011}= \frac{9090\overbrace{9\ldots9}^{n}0909+9091-9091}{9900\underbrace{9\ldots 9}_{n}0099+9901-9901}= \frac{9091\overbrace{0\ldots0}^{n+4}-9091}{9901\underbrace{0\ldots 0}_{n+4}-9901}=\frac{9091}{9901} $$

[rsoldo]

This is Anning's Theorem. Condition in the Theorem is n is odd. I don't know why?! I don't have a counterexample if n is even.

[Rak so dna]

It's just an identity

$\dfrac{101010101_b}{110010011_b}=\dfrac{1+b^2+b^4+b^6+b^8}{1+b+b^4+b^7+b^8}=\dfrac{1+b^2+\frac{b^4(b^n-1)}{b-1}+b^{n+5}+b^{n+7}}{1+b+\frac{b^4(b^n-1)}{b-1}+b^{n+6}+b^{n+7}}=$

$=\dfrac{1+b^2+b^4+\cdots+b^{n+3}+b^{n+5}+b^{n+7}}{1+b+b^4+\cdots+b^{n+3}+b^{n+6}+b^{n+7}}=\dfrac{1010\overbrace{1\ldots1}^{n}0101_b}{1100\underbrace{1\ldots 1}_{n}0011_b}$