Without actually performing the long division, state whether the rational number 129/2^2 x 5^7 x 7^5 have terminating or non-terminating repeating

Question

Without actually performing the long division, state whether the rational number \(\frac{129}{2^2 \times 5^7 \times 7^5}\) have terminating or non-terminating repeating (recurring) decimal expansion.

Answer 1

Given rational number is \(\frac{129}{2^2 \times 5^7 \times 7^5}\)

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

129 = 3 × 43

Denominator = 2² ×5⁷ ×7⁵

129 and 2² ×5⁷ ×7⁵ have no common factors

Therefore, 129 and 2² ×5⁷ ×7⁵ are co-prime.

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

Denominator = 2² ×5⁷ ×7⁵

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

Thus, \(\frac{129}{2^2 \times 5^7 \times 7^5}\) is a non-terminating repeating decimal.