Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Run Euclid's algorithm and reverse it to obtain Bézout coefficients.
- Justify the conclusion "6=−84+3·30" using the stated assumptions.
Before you start
Integer division with remainder.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Express gcd(84,30) as 84a+30b with integers a,b.
Why this math matters
Run Euclid's algorithm and reverse it to obtain Bézout coefficients. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Coefficients may be negative integers.
- The gcd is chosen positive.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Recover a greatest common divisor as a linear combination
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Express gcd(84,30) as 84a+30b with integers a,b.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
84=2·30+24; 30=24+6; 24=4·6
The final nonzero remainder is six.
Work through the mathematics
6=30−24=30−(84−2·30)
Substitute backwards rather than stopping at the gcd.
Check the conclusion
6=−84+3·30
The coefficients −1 and 3 provide an exact certificate that the gcd is generated by the two inputs.
The result
6=−84+3·30
The coefficients −1 and 3 provide an exact certificate that the gcd is generated by the two inputs.
Common mistakes to catch
- A common divisor must divide both numbers.
- Bézout coefficients are generally not unique.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Find coefficients for gcd(21,15).
Show a hint
Reverse 21=15+6 and 15=2·6+3.
Reveal answer and explanation
3=−2·21+3·15
Substitution gives 3=15−2(21−15).
Practice 2
Can 84a+30b equal 7?
Show a hint
Every combination shares a divisor.
Reveal answer and explanation
No
Six divides both terms, so it divides every integer combination but not seven.
Take the idea with you
Use a gcd certificate to decide which total adjustments two fixed-size steps can achieve.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Undo multiplication modulo a prime
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