Euclid's division lemma states that for any positive integers a and b, there exist unique integers q and r such that $a=bq+r$, where r must satisfy:

Question

Accepted Answer

Euclid's division lemma is a fundamental principle in number theory that states for any two positive integers $ a $ and $ b $, there exist unique integers $ q $ (the quotient) and $ r $ (the remainder) such that:

$$ a = bq + r $$

where the remainder $ r $ must satisfy the condition:

$$ 0 \le r < b $$

Let's break down why Option 3 is the correct answer and why the other options are incorrect:

1. **Explanation for Option 3: $ 0 \le r < b $**

- This condition ensures that the remainder $ r $ is non-negative and less than the divisor $ b $. This is crucial because the remainder is the part of $ a $ that is left over after dividing by $ b $, and it cannot be equal to or greater than $ b $, as that would mean $ b $ could still divide into it.
   - The condition $ 0 \le r < b $ guarantees the uniqueness of the quotient $ q $ and the remainder $ r $. If $ r $ were allowed to be equal to or greater than $ b $, it would not be a remainder in the traditional sense, and the division would not be complete.
   - This condition is consistent with the standard definition of division in arithmetic, where the remainder is always less than the divisor.

2. **Why Option 1 is incorrect: $ 1 < r < b $**

- This condition implies that the remainder $ r $ must be greater than 1, which is not always true. For example, if $ a $ is exactly divisible by $ b $, the remainder $ r $ would be 0, which is not allowed by this condition.
   - Additionally, this condition does not allow for the possibility of $ r $ being 0, which is a valid remainder when $ a $ is a multiple of $ b $.

3. **Why Option 2 is incorrect: $ 0 < r \le b $**

- This condition allows $ r $ to be equal to $ b $, which contradicts the definition of a remainder. If $ r $ were equal to $ b $, it would mean that $ b $ could still divide into $ r $, and the division would not be complete.
   - Moreover, this condition excludes the possibility of $ r $ being 0, which is necessary for cases where $ a $ is a multiple of $ b $.

4. **Why Option 4 is incorrect: $ 0 < r < b $**

- This condition implies that the remainder $ r $ must be strictly greater than 0, which excludes the possibility of $ r $ being 0. As mentioned earlier, $ r = 0 $ is a valid remainder when $ a $ is a multiple of $ b $.
   - By excluding 0, this condition fails to account for all possible scenarios in division, particularly when there is no remainder.

In conclusion, Option 3 is the correct answer because it accurately describes the range of possible values for the remainder $ r $ in Euclid's division lemma, ensuring that $ r $ is non-negative and less than the divisor $ b $, which is essential for the uniqueness and completeness of the division.