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§1. Divisibility Basics

AIMO Training · AI-Guided Concept + Practice · based on AoPS Intro to NT Ch 1 + 冯志刚 §1.1
1. Intro
2. Definition
3. Tools Card
4. Guided
5. Practice
6. Reflect
7. Quiz

1. Concept Intro — What does "divide" really mean?

Chat with your AI tutor. It will ask you small questions. Answer honestly — even "I don't know" is fine.

2. Formal Definition

Definition

Divisibility of Integers

For integers $a$ and $b$ with $a \ne 0$, we say $a$ divides $b$ if there exists an integer $k$ such that

$$b = a \cdot k.$$

We write this as $a \mid b$. If no such $k$ exists, we write $a \nmid b$.

Key Properties (性质)
  1. Reflexivity: $a \mid a$ for any nonzero $a$.
  2. Transitivity: If $a \mid b$ and $b \mid c$, then $a \mid c$.
  3. Linearity: If $a \mid b$ and $a \mid c$, then $a \mid (mb + nc)$ for any integers $m, n$.
  4. Bounds: If $a \mid b$ and $b \ne 0$, then $|a| \le |b|$.
Notation Cheat Sheet
$a \mid b$"$a$ divides $b$" — $b$ is a multiple of $a$
$a \nmid b$"$a$ does not divide $b$"
$\gcd(a,b)$ or $(a,b)$greatest common divisor
$\text{lcm}(a,b)$ or $[a,b]$least common multiple

3. Tools Card — Your divisibility toolkit

10 tools you'll use again and again. Click each card to read the full statement, when to use it, and a worked example.

How to use this page: Don't try to memorize. Skim all 10 cards once to see what's there, then come back to specific cards while solving problems. Pay extra attention to the trigger signals — those are the bridge from "reading a problem" to "knowing which tool fits."
0 / 10 marked as known

📋 Quick Lookup Table

ToolOne-line triggerKeywords

Statement
When to use it (trigger signals)
Worked example
Why this tool?

4. Guided Practice (3 problems · ★ basic)

Type your answer. If wrong, the AI tutor will help you.

5. Free Practice (6 problems · ★★ mixed)

AoPS review problems + 冯志刚 basic examples. Stuck? Click "Ask AI" on any problem to chat.

6. Reflection (3 lines)

Research shows: students who reflect after practice retain 2× more. 1-2 sentences each.

7. Checkpoint Quiz (5 questions · unlock next chapter)

Score ≥ 80% (4/5) to pass. Below 60% → recommended review. No AI help here — pure self-check.