Question

Let $a$ & $b$ be integers.

When can $-\frac{1}{a}$ be written as a positive, repeating decimal?

Answer 1

Step 1: $-\frac{1}{a}$ can be written as a positive , repeating decimal for any negative integer such that a don’t have prime factorisation of the form $2^m\times 5^n$.

Step 2: Example 1:

For $a=-12,-\frac{1}{(-12)}=0.8333...$

Step 3: Example 2:

For $a=-7,-\frac{1}{(-7)}=0.\underline{142857}$

Final Answer: $-\frac{1}{a}$ can be written as a positive , repeating decimal for any negative integer such that a don’t have prime factorisation of the form $2^m\times 5^n$.