What can you say about the prime factorisation of the denominator of $27. ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 142857$

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Solution

Let $x = 27. ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 142857$
1000000 x=27142857. $¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 142857$
$\Rightarrow$ 1000000 x -x=271428.57. $¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 142857 - 27. ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 142857$
$\Rightarrow 999999 x = 27142830$
$\Rightarrow x = \frac{27142830}{999999}$
$\Rightarrow x = \frac{27142830}{9 \times 111111}$
$\Rightarrow x = \frac{27142830}{3^{2} \times 111111}$

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What can you say about the prime factorisation of the denominator of 27.¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯142857

Let x=27.¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯1428571000000 x=27142857.¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯142857⇒ 1000000 x -x=271428.57.¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯142857−27.¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯142857⇒999999x=27142830⇒x=27142830999999⇒x=271428309×111111⇒x=2714283032×111111

What can you say about the prime factorisation of the denominator of $27. ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 142857$