State Euclid Division lemma

Answer

**Step-by-Step Solution:**

1. **Understanding the Terms:**
- Let \( a \) and \( b \) be two positive integers.
- Here, \( a \) is known as the dividend, and \( b \) is the divisor.

2. **Statement of Euclid's Division Lemma:**
- According to Euclid's Division Lemma, for any two positive integers \( a \) and \( b \), there exist unique integers \( q \) (the quotient) and \( r \) (the remainder) such that:
\[ a = bq + r \]
- This equation represents the relationship between the dividend, divisor, quotient, and remainder.

3. **Condition on the Remainder:**
- The remainder \( r \) must satisfy the condition:
\[ 0 \leq r < b \]
- This means that \( r \) can take values starting from 0 up to but not including \( b \).

4. **Comparison with Division Algorithm:**
- This lemma can be compared to the division algorithm, which states:
\[ \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \]
- In this context, \( a \) is the dividend, \( b \) is the divisor, \( q \) is the quotient, and \( r \) is the remainder.

5. **Example for Clarity:**
- If we take \( b = 3 \), then the possible values of \( r \) can be:
\[ r = 0, 1, 2 \]
- The value 3 is not included because \( r \) must be less than \( b \).