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 $\frac{13}{280}$

Answer 1

Terminating decimals:

$\frac{p}{q}$ is terminating if

1. p and q are co-prime.
2. q is of the form $2^n{5}^m$.
   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.

Given data:

Given number is $\frac{13}{280}$

Explanation:

Checking co-primes:

1. 13 is a prime number (As 13 is divisible by 1 and 13 only)
2. Factors of 280 are $280=2\times 2\times 2\times 5\times 7$
3. Then, 13 and 280 have no common factors.
4. So, 13 and 280 are co-primes.

Denominator:

1. Here the denominator $280=2\times 2\times 2\times 5\times 7=2^3\times 5^1\times 7^1$
2. Which is not in the form $2^n{5}^m$

Conclusion:

Hence, $\frac{13}{280}$is a non-terminating repeating decimal number.