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GCD and LCM (ggT and kgV)

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

How much are \( \frac{1}{4} \) and \( \frac{1}{6} \) of a pizza together? You need a common denominator. You learn the LCM (kgV) for the denominator and the GCD (ggT) for simplifying.
📄 Open worksheet (PDF, German)

The video

In the video we cut two pizzas into pieces of the same size. New cuts are purple. When we simplify, purple cuts disappear again.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

Contents of the video

  1. 0:00 Different pieces
  2. 0:29 The least common multiple
  3. 1:01 The greatest common divisor
  4. 1:30 Simplifying with the GCD
  5. 1:52 Careful, mistake!
  6. 2:14 Try it yourself
  7. 2:52 Summary

The most important points

The LCM (das kgV)

Multiples of 4: 4, 8, 12, 16, … Multiples of 6: 6, 12, 18, … The least common multiple is \( \text{kgV}(4, 6) = 12 \). It is the best common denominator: \( \frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \).

12 is in both rows.
12 is in both rows.

The GCD (der ggT)

Divisors of 12: 1, 2, 3, 4, 6, 12. Divisors of 18: 1, 2, 3, 6, 9, 18. The greatest common divisor is \( \text{ggT}(12, 18) = 6 \).

12/18 of a pizza and the divisors of 12 and 18.
\( \frac{12}{18} \) of a pizza and the divisors of 12 and 18.

With the GCD you simplify in one step: \( \frac{12}{18} = \frac{12 : 6}{18 : 6} = \frac{2}{3} \). Do not mix them up: for the denominator you need the LCM, for simplifying the GCD.

Every 6 small pieces become one big piece: two thirds.
Every 6 small pieces become one big piece: two thirds.

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

The worksheet – Page 1
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The worksheet – Page 2
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The worksheet – Page 3
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The worksheet – Page 4
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Show the worksheet solutions
The worksheet – Page 5
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Try it yourself

Task

a) \( \frac{1}{6} + \frac{1}{8} \) b) Simplify \( \frac{16}{24} \) in one step.

Show solution

a) \( \text{kgV}(6, 8) = 24 \): \( \frac{4}{24} + \frac{3}{24} = \frac{7}{24} \) b) \( \text{ggT}(16, 24) = 8 \): \( \frac{16}{24} = \frac{2}{3} \)

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: ggT and kgV (the LCM as common denominator, simplifying with the GCD). Give me five short tasks, one after the other. Always wait for my answer, then explain briefly. Write in simple German and put an English translation in brackets after each sentence.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.