Euclid's division Lemma states that for two positive integers a and b, there exist unique integers q and r such that `a=bq+r` where r must satisfy.

Options

A. `1 lt r ltb`
B. `0 lt r leb`
C. `0 le r lt b`
D. `0 lt r lt b`

Answer

To solve the question regarding Euclid's division lemma, we need to understand the statement and the conditions it imposes.

### Step-by-Step Solution:

1. **Understanding Euclid's Division Lemma**:
Euclid's division lemma states that for any two positive integers \( a \) and \( b \), there exist unique integers \( q \) (the quotient) and \( r \) (the remainder) such that:
\[ a = bq + r \]

2. **Identifying the Conditions for \( r \)**:
The remainder \( r \) must satisfy two conditions:
- \( r \) must be greater than or equal to 0.
- \( r \) must be less than \( b \).

Therefore, we can express this condition mathematically as:
\[ 0 \leq r < b \]

3. **Conclusion**:
Thus, the complete statement of Euclid's division lemma can be summarized as:
For any two positive integers \( a \) and \( b \), there exist unique integers \( q \) and \( r \) such that:
\[ a = bq + r \quad \text{where} \quad 0 \leq r < b \]

### Final Answer:
The conditions that \( r \) must satisfy are:
- \( r \geq 0 \)
- \( r < b \)