medium difficulty6 min

Iterative division for HCF

Iterative division for HCF is a step-by-step method where the larger number is divided by the smaller number, then the divisor and remainder are used again until the remainder becomes 0. The last non-zero remainder is the HCF.

In one line: Keep the divisor, drop the old dividend, last non-zero wins.

Practice This Concept

Understand it

The HCF does not change when the larger number is replaced by the smaller number and the remainder. That is why the same logic can be used again and again, making the numbers smaller at every step. The method is fast and neat in exam work.

Example

For 252 and 198: 252 = 198 × 1 + 54, 198 = 54 × 3 + 36, 54 = 36 × 1 + 18, 36 = 18 × 2 + 0. So the HCF is 18.

Formula and rule

If a = bq + r, then HCF(a,b) = HCF(b,r).

See it worked out

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

Worked example

1Find the HCF of 252 and 198 by repeated division: 252 = 198 × 1 + 54, 198 = 54 × 3 + 36, 54 = 36 × 1 + 18, 36 = 18 × 2 + 0.
2Since the last non-zero remainder is 18, the HCF is 18.

Method you can reuse

1Divide the larger number by the smaller number.
2Write the remainder.
3Replace the pair by divisor and remainder.
4Continue until remainder is 0.
5The last non-zero remainder is the HCF.

Visual tip

A vertical division ladder with arrows from one step to the next is very useful for this method.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

In repeated division, why do we replace the pair with divisor and remainder?

Think first, then show answer

Because the HCF stays the same, and the numbers become smaller until the last non-zero remainder is reached.

Common wrong answer

Taking 36 as the HCF in the worked example.

Why it loses marks: 36 is not the last non-zero remainder.

Don't mix this up

Students stop after the first remainder or take the quotient instead of the last non-zero remainder.

Exam tip

Write each division clearly and stop only when the remainder becomes 0. The last non-zero remainder is the answer.

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

Use repeated division to find the HCF of two numbers, or explain why the method works.

1
Enough for 1 mark
Iterative division for HCF is a step-by-step method where the larger number is divided by the smaller number, then the divisor and remainder are used again until the remainder becomes 0. The last non-zero remainder is the HCF.
2
Add for 2 marks
Example: For 252 and 198: 252 = 198 × 1 + 54, 198 = 54 × 3 + 36, 54 = 36 × 1 + 18, 36 = 18 × 2 + 0. So the HCF is 18.
3
Add for 3 marks
The HCF does not change when the larger number is replaced by the smaller number and the remainder.

MCQ Quiz

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