Question

Without actually performing the long division, state whether the rational number 64/455 have terminating or non-terminating repeating (recurring) decimal expansion.

Answer 1

Given rational number is 64/455

p/q is terminating if

a) p and q are co-prime &

b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.

Firstly, we check co-prime

64 = 2⁶

455 = 5 × 7 × 13

64 and 455 have no common factors

Therefore, 64 and 455 are co-prime.

Now, we have to check that q is in the form of 2ⁿ5^m

455 = 5 × 7 × 13

So, denominator is not of the form 2ⁿ5^m

Thus, 64455 is a non-terminating repeating decimal.