Euclid's Division Lemma
Learn how every division can be written in a structured form and why that helps in HCF problems.
- 2 min read
- Aligned to CBSE 2025–26 syllabus
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- Updated Sept 2026
Use this map for a first read, then return to the checks and formulas during revision.
- 1 key formulas
Statement
Given positive integers a and b, there exist unique whole numbers q and r such that the dividend can be written as divisor x quotient + remainder, with the remainder always smaller than the divisor.
The quotient q tells us how many complete groups of b fit into a, and the remainder r is what is left over.
How to Use It
To find HCF, keep rewriting the dividend and divisor using the lemma until the remainder becomes zero.
The last non-zero remainder's divisor is the HCF.
Need more chapter practice?
Move to the chapter practice section to test this concept with board-style questions.
Go to PracticeTopic Q&A
Why must the remainder be smaller than the divisor?
If the remainder were equal to or bigger than the divisor, you could subtract one more divisor and increase the quotient, so the division would not be complete.
Is the quotient always unique?
Yes. For fixed positive integers a and b, the quotient and remainder in Euclid's Division Lemma are unique.