Question
Without actually performing the long division, state whether state whether the following rational numbers will have a terminating decimal expansion or a non terminating repeating decimal expansion.
$\frac{77}{210}$
$\frac{77}{210}$
✓
Answer
The given number is $\frac{77}{210}$ and $HCF(77, 210) = 7.$
$\therefore\ \frac{77}{210}=\frac{77\div7}{210\div7}=\frac{11}{30}$
Here, $\frac{11}{30}$ is in its simplest form.
Here, $\frac{77}{210}$ is in its simplest form.
Now, $30 = 2 \times 3 \times 5$ is not of the form $2^m \times 5^n.$
So, the given number has a non-terminating repeating decimal expansion.
$\therefore\ \frac{77}{210}=\frac{77\div7}{210\div7}=\frac{11}{30}$
Here, $\frac{11}{30}$ is in its simplest form.
Here, $\frac{77}{210}$ is in its simplest form.
Now, $30 = 2 \times 3 \times 5$ is not of the form $2^m \times 5^n.$
So, the given number has a non-terminating repeating decimal expansion.
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