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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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