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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.

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Student-friendly 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.

How to write this in exams

  1. 1

    Start with the exact idea

    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

    Then show how to use it

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

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete answer

    Writing 157 = 12 x 12 + 13 is wrong because the remainder 13 is not smaller than the divisor 12.

Definition

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.

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.

Rule to remember

a = bq + r, where 0 <= r < b.

Memory hook

Remainder stays small: always smaller than the divisor.

Examples and method

Worked example

Write 157 = 12 x 13 + 1. Here 157 is the dividend, 12 is the divisor, 13 is the quotient, and 1 is the remainder. Since 1 is smaller than 12, the form is correct.

Method to apply

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

Diagram support

A simple division ladder with boxes for dividend, divisor, quotient, and remainder helps students check the form quickly.

How CBSE asks it

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

Avoid common mistakes

Common confusion

Students often write a remainder equal to or larger than the divisor, or they forget that the remainder cannot be negative.

Common wrong answer

Writing 157 = 12 x 12 + 13 is wrong because the remainder 13 is not smaller than the divisor 12.

Exam tip

Always write the condition 0 <= r < b after the division form. Many CBSE answers expect both the equation and the remainder condition.

Quick check

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

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

Answer writing and exam use

1-mark answer

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-mark answer

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. a = bq + r, where 0 <= r < b. When 157 is divided by 12, we can write 157 = 12 x 13 + 1, so the quotient is 13 and the remainder is 1.

3-mark answer

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. a = bq + r, where 0 <= r < b. Write 157 = 12 x 13 + 1. Here 157 is the dividend, 12 is the divisor, 13 is the quotient, and 1 is the remainder. Since 1 is smaller than 12, the form is correct. Find quotient and remainder for a given division, or verify whether a written division form is valid. Writing 157 = 12 x 12 + 13 is wrong because the remainder 13 is not smaller than the divisor 12.
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