Question

Write the statement of Euclid's Division Lemma.

Hint:

Remember: Euclid's Division Lemma is based on the idea that Dividend = Divisor × Quotient + Remainder, where the remainder is always less than the divisor.

Answer 1

Step 1: State the lemma in words.}
Euclid's Division Lemma states that for any two positive integers, there exist unique integers \( q \) and \( r \) such that a given integer can be expressed in the form of divisor multiplied by quotient plus remainder.
Step 2: Write the mathematical form.}
If \( a \) and \( b \) are two positive integers, then there exist unique whole numbers \( q \) and \( r \) such that \[ a = bq + r \] where \( 0 \leq r<b \).
Step 3: Explain the meaning of symbols.}
Here, \( a \) is the dividend, \( b \) is the divisor, \( q \) is the quotient, and \( r \) is the remainder.
Step 4: Mention the condition on remainder.}
The remainder must always be non-negative and smaller than the divisor. This condition makes the statement complete and correct.
Step 5: Final statement.}
Therefore, the statement of Euclid's Division Lemma is: For any two positive integers \( a \) and \( b \), there exist unique integers \( q \) and \( r \) satisfying \[ a = bq + r, \quad 0 \leq r<b \]