Which of the following rational numbers have the terminating decimalrepresentation?
(i)3/5             (ii)7/20                   (iii)2/13
(iv) 27/40        (v)133/125            (vi)23/7
[Making use of the result that a rational number p/q where p and q have nocommon factor(s) will have a terminating representation if and only if theprime factors of q are 2's or 5's or both.

(i) In , the denominator is 5.

Clearly, 3 & 5 do not have any common factors.

Also, the prime factor of 5 is 5.

Hence 3/5 has a terminating decimal representation.

(ii) In , the denominator is 20.

Clearly, 7 & 20 do not have any common factors.

Also Prime factors of 20 are = 4 × 5 = 2 ² × 5.
Which are both 2's and 5's.

Hence 7/20 has a terminating decimal representation.

(iii) In , the denominator is 13.

Clearly, 2 & 13 do not have any common factors.

Prime factor of 13 is 13.

Which is neither 2's nor 5's.

Hence 2/13 has non-terminating decimal representation.

(iv) In , the denominator is 40.

Clearly, 27 & 40 do not have any common factors.

Also prime factors of 40 are 2 ³ × 5.
Which are both 2's and 5's.

Hence 27/40 has a terminating decimal representation.

(v) In , the denominator is 125.

Clearly, 133 & 125 do not have any common factors.

Also prime factors of 125 are 5 ³
Which are 5's.

Hence 133/125 has a terminating decimal representation.

(vi) In , the denominator is 7.

The prime factor of 7 is 7.

Prime factors are neither 2’s nor 5’s.

Hence 23/7 has a non-terminating decimal representation.