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.
For integers a and b (with b ≠ 0), we say "b divides a" (written b ∣ a) if:
There exists an integer k such that a = bk.
Direct proofs of divisibility statements always come back to this definition.
Example:
Prove: If a is divisible by 6, then a is divisible by 3.
Assume 6 ∣ a.
Then there exists an integer k such that a = 6k.
But 6k = 3(2k), so a = 3(2k).
Let m = 2k (an integer), then a = 3m, so 3 ∣ a.
Typical patterns:
You almost always: start with a = dk, do algebra, and show the result equals d × (integer) or some other divisor × (integer).
TL;DR — key idea
Divisibility proofs unwrap "d ∣ n" into n = dk, manipulate that equation, and then rewrap it to show divisibility by another integer.
Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.
Prove one of the following using a direct proof: 1. If n is divisible by 4, then n is divisible by 2. 2. If a and b are both divisible by 5, then a + b is divisible by 5. Be explicit about where you use the definition "d divides n ⇔ n = dk for some integer k".
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.