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Without Actual Division Show that Each of the Following Rational Numbers is a Non-terminating Repeating Decimal. (I)`77/210` - Mathematics

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Without actual division show that each of the following rational numbers is a non-terminating repeating decimal. 

(i)`77/210`

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Solution

`77/210 =( 77 ÷7)/(210 ÷7) = 11/30 = 11/(2 ×3 ×5)`
We know 2, 3 or 5 is not a factor of 11, so `11/30` is in its simplest form.
Moreover, (2 × 3 × 7) ≠ (2m × 5n
Hence, the given rational is non-terminating repeating decimal.

Concept: Euclid’s Division Lemma
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