For two positive integers a and b, there exist unique integers q and r such that a = bq + r, then r is :

Question

Accepted Answer

To understand why Option 3 is the correct answer, we need to delve into the division algorithm, which is a fundamental concept in number theory. The division algorithm states that for any two integers $ a $ and $ b $ (where $ b > 0 $), there exist unique integers $ q $ (the quotient) and $ r $ (the remainder) such that:

$$ a = bq + r $$

where the remainder $ r $ satisfies the condition:

$$ 0 \le r < b $$

This condition ensures that the remainder $ r $ is non-negative and strictly less than the divisor $ b $. Let's break down why Option 3 is correct and why the other options are not:

1. **Option 3: $ 0 \le r < b $**
   - This option correctly reflects the condition of the remainder in the division algorithm. The remainder $ r $ can be zero, which is why the inequality includes $ 0 \le r $. Additionally, $ r $ must be less than $ b $, ensuring that $ r $ is not equal to or greater than $ b $. This is the precise definition of the remainder in the context of integer division.

2. **Option 1: $ 1 < r < b $**
   - This option is incorrect because it implies that the remainder $ r $ must be greater than 1, which is not always the case. For example, if $ a = 5 $ and $ b = 3 $, then $ r = 2 $, which satisfies $ 0 \le r < b $ but not $ 1 < r < b $. Additionally, if $ a = 3 $ and $ b = 3 $, then $ r = 0 $, which is valid under the correct condition but not under this option.

3. **Option 2: $ 0 < r \le b $**
   - This option is incorrect because it allows for the possibility that $ r = b $, which contradicts the requirement that $ r $ must be less than $ b $. For instance, if $ a = 6 $ and $ b = 3 $, then $ r = 0 $, which is valid under the correct condition but not under this option.

4. **Option 4: $ 0 < r < b $**
   - This option is incorrect because it excludes the possibility of $ r = 0 $. The division algorithm allows for $ r $ to be zero, as seen in cases where $ a $ is a multiple of $ b $. For example, if $ a = 6 $ and $ b = 3 $, then $ r = 0 $, which is valid under the correct condition but not under this option.

In conclusion, Option 3 accurately captures the conditions of the division algorithm, making it the correct choice. The other options fail to account for all possible values of $ r $ as defined by the division algorithm.