Without actual division , show that the following rational number is non-terminating repeating decimal :

Foundation

Mathematics Foundation

HCF and LCM

Properties of Real Number

Recurring Decimal Number

Question

Without actual division , show that the following rational number is non-terminating repeating decimal : $\frac{64}{,} 455$

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Given rational number is

\frac{64}{455}

Prime factorization of the denominator of the given rational number is

455 = 5 × 7 × 13

If the prime factorization of the denominator of any rational number is of the form 2^m × 5ⁿ where m and n are non-negative integers, then the decimal expansion of the given number is terminating else non-terminating repeating

(5 × 7 × 13) is not of the form (2^m× 5ⁿ)
Hence given fraction has non -terminating repeating decimal expansion.