Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Solve simultaneous congruences and identify the repetition period.
- Justify the conclusion "x≡8 mod 15" using the stated assumptions.
Before you start
Remainders and substitution.
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
A marker occurs at x≡2 mod 3 and x≡3 mod 5. Find all integer positions.
Why this math matters
Solve simultaneous congruences and identify the repetition period. 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:
- The moduli three and five are coprime.
- Negative as well as positive integer positions are allowed.
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.
Combine two independent remainder schedules
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A marker occurs at x≡2 mod 3 and x≡3 mod 5. Find all integer positions.
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
x=2+3k
The first schedule restricts x to one arithmetic progression.
Work through the mathematics
2+3k≡3 mod 5 ⇒ 3k≡1 ⇒ k≡2 mod 5
Multiplication by the inverse of three gives the allowed k values.
Check the conclusion
x≡8 mod 15
Writing k=2+5m gives x=8+15m; coprime moduli give one class modulo their product.
The result
x≡8 mod 15
Writing k=2+5m gives x=8+15m; coprime moduli give one class modulo their product.
Common mistakes to catch
- Noncoprime congruences need a compatibility check.
- One representative is not the complete solution set.
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
Solve x≡1 mod 2 and x≡2 mod 3.
Show a hint
List odd residues modulo six.
Reveal answer and explanation
x≡5 mod 6
Five is odd and leaves remainder two on division by three.
Practice 2
Can x be even and also 1 mod 4?
Show a hint
Reduce the second condition modulo two.
Reveal answer and explanation
No
Every integer congruent to one modulo four is odd.
Take the idea with you
Combine two periodic maintenance schedules and explain the least common repetition interval.
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: Reuse elimination with an LU factorization
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