Euclid division lemma
If one integer a is divided by another positive integer b, we can write a = bq + r, where q is the quotient and r is the remainder, with 0 <= r < b.
Practice This ConceptMain explanation
Teacher explanation
This lemma gives the exact form of division with remainder. It tells us that the remainder is not free to take any value; it must stay smaller than the divisor and cannot be negative. That strict form is useful in HCF methods and in proof-based questions.
Example
When 157 is divided by 12, we can write 157 = 12 x 13 + 1, so the quotient is 13 and the remainder is 1.
Simple analogy
Remainder stays small: always smaller than the divisor.
Common confusion
Students often write a remainder equal to or larger than the divisor, or they forget that the remainder cannot be negative.
Exam tip
Always write the condition 0 <= r < b after the division form. Many CBSE answers expect both the equation and the remainder condition.
Answer writing and exam use
1-mark use
Write the exact meaning of euclid division lemma in one clean line.
2-mark use
Define euclid division lemma and add one example or condition.
3-mark use
Explain euclid division lemma, show the method or example, and mention the common mistake.
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