Theorem: Let x = p/q be a rational number, such that the prime factorisation of q is of the form 2ⁿ5ᵐ, where n, m are non-negative integers. Then, x has a decimal expansion which terminates
(i) 133/125
Factorise the denominator, we get 125 = 5 × 5 × 5 × 5 = 5⁴
So, the denominator is in the form of 5ᵐ, so, 133/125 is terminating
(ii) 17/8
Factorise the denominator, we get 8 = 2 × 2 × 2 = 2³
So, the denominator is in the form of 2ⁿ, so, 17/8 is terminating
(iii) 64/455
Factorise the denominator, we get 455 = 5 × 7 × 13
So, the denominator is not in the form of 2ⁿ5ᵐ, so, 64/455 is not terminating
(iv) 15/1600
Factorise the denominator, we get 1600 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 = 2⁶5²
So, the denominator is in the form of 2ⁿ5ᵐ, so, 15/1600 is terminating
(v) 29/343
Factorise the denominator, we get 343 = 7 × 7 × 7 = 7³
So, the denominator is not in the form of 2ⁿ5ᵐ, so, 29/343 is not terminating
(vi) 23/2352
Here, the denominator is in the form of 2ⁿ5ᵐ, so, 23/2352 is terminating
(vii) 129/225775
Here, the denominator is not in the form of 2ⁿ5ᵐ, so, 129/225775 is not terminating
(viii) 6/15
Divide numerator and denominator both by 3 we get 2/5
So, the denominator is in the form of 5ᵐ, so, 6/15 is terminating
(ix) 35/50
Divide numerator and denominator both by 5 we get 7/10
Factorise the denominator, we get 10 = 2 × 5
So, the denominator is in the form of 2ⁿ5ᵐ, so, 35/50 is terminating
(x) 77/210
Divide numerator and denominator both by 7 we get 11/30
Factorise the denominator, we get 30 = 2 × 3 × 5
So, the denominator is not in the form of 2ⁿ5ᵐ, so 77/210 is not terminating