Without doing actual division , show that the following rational number has a non-terminating repeating decimal expansion :

Foundation

Mathematics Foundation

Properties of Real Number

Recurring Decimal Number

Linear Equations in One Variable

Question

Without doing actual division , show that the following rational number has a non-terminating repeating decimal expansion : $\frac{29}{343}$

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The given number is

\frac{29}{343}

Now,

$\frac{29}{343} = \frac{29}{7 \times 7 \times 7} = \frac{29}{7^{3}}$
7 is not a factor of 29, so the given fraction is in its simplest form.

Here, the denominator is not in the form

(2 m \times 5 n)

∴ (7 3) \neq (2 m \times 5 n)

Hence, given rational number has a non-terminating repeating decimal expansion.