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Question

Without actually performing the long division, state whether the following rational number will have the terminating decimal expansion or a non-terminating repeating decimal expansion 31253×52×22


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Solution

Definition:

pqis terminating if

  1. p and q are co-prime
  2. q is of the form 2n×5m

Co-primes:

A pair of numbers is called co-primes if those numbers have only one common factor, namely 1.

Example:

3,5 are co-primes as they have one common factor which is 1.

Explanation:

Checking Co-primes:

  1. Factors of 3125=5×5×5×5×5
  2. Factors of 3×52×22=3×5×5×2×11
  3. 3125and3×52×22 have common factors 5.
  4. So, 3125and3×52×22 are not co-primes.

Denominator:

  1. Here, the denominator 3×52×22 cannot be written in the form 2n5m.

Conclusion:

Hence, 31253×52×22 is a non-terminating repeating decimal number.


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