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State true or false:
The fraction $\frac{77}{210}$ will have a non-terminating repeating decimal expansion.
(Solve without the long division method)

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True

False

answer is B.

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Detailed Solution

The given fraction is

\frac{77}{210}

.
The given fraction in its simplest form is,

\frac{77}{210} = \frac{7 \times 11}{2 \times 3 \times 5 \times 7}

\Rightarrow \frac{77}{210} = \frac{11}{2 \times 3 \times 5}

\Rightarrow \frac{77}{210} = \frac{11}{30}

Factorization of

30 = 2 \times 3 \times 5

Observe that the factors of 30 cannot be expressed in the form of

2^{m} \times 5^{n}

where m and n are non negative integers.
Therefore, it is true that the fraction

\frac{77}{210}

has got a non-terminating repeating decimal expansion.
Hence, option 1 is correct.

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