Question
Without actual division, show that the following rational numbers is a non-terminating repeating decimal:
$\frac{129}{\big(2^2\times5^7\times7^5\big)}$

Answer

$\frac{129}{\big(2^2\times5^7\times7^5\big)}$We know $2, 5$ or $7$ is not a factor of $129$, so it is in its simplest form.
Moreover, $(2^2 \times 5^7 \times 7^5) \neq (2^m \times 5^n)$
Hence, the given rational is non-terminating repeating decimal.

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