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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 18


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Solution

Definition:

pqis terminating if

  1. p and q are co-prime.
  2. q is of the form 2nand5m
    Where n and m are non-negative integers.

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.

623e0a0588db7c968919e44b623e0a0588db7c968919e44b

Explanation:

Checking co-prime:

  1. 1 is a prime number.
  2. Factors of 8 are 8=2×2×2
  3. 1 and 8 has no common factors.
  4. So, 1 and 8 are co-primes.

Denominator:

  1. Here 8=23×50 (As,50=1)
  2. Which is in the form 2n5m where, n=3 and m=0

Conclusion:

Hence, 18is a terminating decimal number.


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