Direct Proof Practice Problems

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.

10–20 min focusProof-first mindset

Best flow: read → think on paper → write a short explanation → refine with feedback.

Reading

Core explanation

Here are some classic direct-proof style problems. You don't need tricks—just definitions and careful algebra/logic.

Try to prove each of the following:

If n is an even integer, then n² + 6 is even.
If a and b are odd integers, then a² + b² is even.
If n is divisible by 3 and 4, then n is divisible by 12.
If x ≥ 1, then x³ ≥ x.
If m and n are integers and m is divisible by n, then m² is divisible by n.

Focus on:

Translating definitions (even/odd, divisibility).
Doing clean algebra.
Ending with a statement that directly matches the definition of what you’re trying to prove.

TL;DR — key idea

These problems are pure reps: use the same direct-proof patterns (assume, translate definitions, do algebra, translate back, conclude) until the process feels natural.

Try these in your notebook

Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.

1
Choose TWO of the practice statements in the reading and write full direct proofs for them. After writing, check: - Did you clearly state your assumptions at the start? - Did you explicitly use the relevant definitions (even, odd, divides, ≥, etc.)? - Does the last line of your proof exactly match the statement you were trying to prove?

Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.

Check your understanding

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.