Lesson subsection
Read the explanation, try the on-paper prompts, then explain the idea in your own words. Use AI feedback as a mentor, not a shortcut.
Best flow: read → think on paper → write a short explanation → refine with feedback.
In algebra and number theory, counterexamples often appear when intuition fails or when assumptions are missing.
Common examples:
Cancellation law failure in modular arithmetic
2x ≡ 2y (mod 4) does NOT imply x ≡ y (mod 4).
Failure of exponent rules
(a + b)² ≠ a² + b² in general.
Failure of unique factorization in some number systems
In ℤ[√−5], 6 = 2·3 = (1 + √−5)(1 − √−5).
Divisibility traps
"If a ∣ bc, then a ∣ b or a ∣ c" is false.
Example counterexample:
2 divides 4·3, but 2 does not divide 3.
Counterexamples prevent incorrect overgeneralizations and refine intuition.
TL;DR — key idea
Algebra and number theory contain many tempting but false generalizations—counterexamples show where intuition must be corrected.
Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.
Provide counterexamples to TWO of the following: 1. "If a ∣ bc, then a ∣ b." 2. "If x² = y², then x = y." 3. "If a ≡ b (mod n), then a² ≡ b² (mod n) always holds." (Hint: does it ALWAYS hold?) Explain what structural assumption fails in each example.
Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.
In 3–6 sentences, explain the core idea of this subsection as if you were teaching a friend who hasn’t seen it. Focus on the logic, not just the final statements.
AI is optional. Use it to spot gaps and sharpen your wording, not to replace your own thinking.