Question

What can you say about the prime factorisation of the denominators of the following rationals :
$34.\overline{5678}$

Answer 1

$34.\overline{5678}$

Let $x=34.\overline{5678}=\mathrm{34.56785678....}$

Now,

$10000x=\mathrm{345678.5678.....}$

Noe $10000x-x=345678.\overline{5678}-34.\overline{5678}$

$9999x=345644$

$x=\frac{345644}{9999}$

Now prime factorization of denominator of $x$ is

$x=\frac{345644}{(3{)}^2\times 11\times 101}$