C
CraftExam
medium difficulty8 min

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.

In one line: Remainder stays small: always smaller than the divisor.

Practice This Concept

Understand it

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 × 13 + 1, so the quotient is 13 and the remainder is 1.

Formula and rule

a = bq + r, where 0 ≤ r < b.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Write 157 = 12 × 13 + 1.

  1. 1Here 157 is the dividend, 12 is the divisor, 13 is the quotient, and 1 is the remainder.
  2. 2Since 1 is smaller than 12, the form is correct.

Method you can reuse

  1. 1Divide a by b.
  2. 2Write a = bq + r.
  3. 3Check that r is not negative.
  4. 4Check that r is smaller than b.
  5. 5If both conditions hold, the form is correct.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

In the division of 157 by 12, what must be true about the remainder?

Think first, then show answer
The remainder must be less than 12 and cannot be negative, so it can only lie between 0 and 11.

Common wrong answer

Writing 157 = 12 × 12 + 13.

Why it loses marks: The remainder 13 is not smaller than the divisor 12.

Don't mix this up

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.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Find quotient and remainder for a given division, or verify whether a written division form is valid.
  1. 1

    Enough for 1 mark

    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.

  2. 2

    Add for 2 marks

    Example: When 157 is divided by 12, we can write 157 = 12 × 13 + 1, so the quotient is 13 and the remainder is 1.

  3. 3

    Add for 3 marks

    This lemma gives the exact form of division with remainder.

MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results
Practice This Concept

Help improve this page

Found something confusing, incorrect, or missing?

Practice This Concept