Question

What are the different possible decimal expansions that you can expect when you divide any integer with $3$?

• A. $0,0.33\dots ,\mathrm{0.666....}$
• B. $0,0.3,0.6$
• C. $0,0.33\dots ,0.5$
• D. $0.33\dots ,0.5$

Answer 1

The correct option is A $0,0.33\dots ,\mathrm{0.666....}$

1. When the integer is divisible by $3,$ the decimal expansion will terminate in $0$ i.e. the integer is divisible by $3.$

2. Let's divide $1$ by $3.$ We get:


3. Let's divide $2$ by $3.$ We get:


Hence, if any integer is divided by $3,$ there are $3$ possibilities:
1. The decimal expansion will terminate.
2. The decimal expansion will have $3$ repeating.
3. The decimal expansion will have $6$ repeating.

Hence, option A is correct.