#### Without actually performing the king division, State whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

(i) 13/3125

(ii) 17/8

(iii) 23/75

(iv) 6/15

(v) 1258/625

(vi) 77/210

**Solution:**

We know that, if the denominator of a fraction has only 2 or 5 or both factors, it is a terminating decimal otherwise it is non-terminating repeating decimals.

**(i)** **13/3125**

3125 = 5 × 5 × 5 × 5 × 5

Prime factor of 3125 = 5, 5, 5, 5, 5 [i.e., in the form of 2^{n}, 5^{n}]

It is a terminating decimal.

**(ii) 17/8**

8 = 2 × 2 × 2

Prime factor of 8 = 2, 2, 2 [i.e., in the form of 2^{n}, 5^{n}]

It is a terminating decimal.

**(iii) **23/75

75 = 3 × 5 × 5

Prime factor of 75 = 3, 5, 5

It is a non-terminating repeating decimal.

**(iv) 6/15**

Let us divide both numerator and denominator by 3

6/15 = (6 **÷ **3) / (15 **÷ **3)

= 2/5

Since the denominator is 5.

It is a terminating decimal.

**(v) **1258/625

625 = 5 × 5 × 5 × 5

Prime factor of 625 = 5, 5, 5, 5 [i.e., in the form of 2^{n}, 5^{n}]

It is a terminating decimal.

**(vi) 77/210**

Let us divide both numerator and denominator by 7

77/210 = (77 **÷ **7) / (210 **÷ **7)

= 11/30

30 = 2 × 3 × 5

Prime factor of 30 = 2, 3, 5

It is a non-terminating repeating decimal.

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