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Question

Without actual division, show that each of the following rational numbers is a nonterminating repeating decimal.

(i) 11(23×3) (ii) 73(22×33×5) (iii) 129(22×53×72) (iv) 935(v) 77210 (vi) 32147 (vii) 29343 (viii) 64455

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Solution

(i) 1123×3

We know either 2 or 3 is not a factor of 11, so it is in its simplest form.

Moreover, (23×3)(2m×5n)

Hence, the given rational is non-terminating repeating decimal.

(ii) We know 2, 3 or 5 is not a factor of 73, so it is in its simplest form.

Moreover, (22×33×5)(2m×5n)

Hence, the given rational is non-terminating repeating decimal.

(iii) We know 2, 5 or 7 is not a factor of 129, so it is in its simplest form.

Moreover, (23×57×75)(2m×5n)

Hence, the given rational is non-terminating repeating decimal.

(iv) 935 = 97×5

We know either 5 or 7 is not a factor of 9, so it is in its simplest form. Moreover, (5×7)(2m×5n)

Hence, the given rational is non-terminating repeating decimal.

(v) 77210 = 777×30 = 1130

112×3×5

We know 2, 3 or 5 is not a factor of 11, so 1130 is in its simplest form.

Moreover, (2×3×5)(2m×5n)

Hence, the given rational is non-terminating repeating decimal.

(vi) We know either 3 or 7 is not a factor of 32, so it is in its simplest form.

Moreover, (3×72)(2m×5n)

Hence, the given rational is non-terminating repeating decimal.

(vii) We know 7 is not a factor of 29, so it is in its simplest form.

Moreover, 73(2m×5n)

Hence, the given rational is non-terminating repeating decimal.

(viii)64455=645×7×13

We know 5, 7 or 13 is not a factor of 64, so it is in its simplest form.

Moreover, (5×7×13)(2m×5n)

Hence, the given rational is non-terminating repeating decimal.


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