AMC 8 Master Workbook · 20 Essential Problems

① Number Theory Number Theory

Key Facts to Memorize:
• Divisibility rules: 2(even), 3(digit sum÷3), 4(last 2 digits÷4), 5(ends 0 or 5), 6(÷2 and ÷3), 9(digit sum÷9)
• GCD × LCM = product of two numbers
• Prime numbers up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
• Number of factors of $p^a q^b$ = $(a+1)(b+1)$

LCM(a, b) = ab ÷ GCD(a, b)

Example

How many positive divisors does 60 have?
60 = 2² × 3 × 5 → factors = (2+1)(1+1)(1+1) = 12

Answer: 12

② Algebra & Arithmetic Algebra

Key Formulas:
• Percent change: $\frac{\text{new}-\text{old}}{\text{old}} \times 100\%$
• Average: $\text{Sum} \div \text{Count}$
• Distance = Rate × Time
• Simple Interest: $I = P \cdot r \cdot t$

If a : b = c : d → a × d = b × c

Example

A price increases from \$80 to \$100. What is the % increase?

Answer: (100−80)/80 × 100% = 25%

③ Geometry Geometry

Must-Know Formulas:
• Area of triangle: $\frac{1}{2}bh$
• Area of circle: $\pi r^2$; Circumference: $2\pi r$
• Pythagorean triples: (3,4,5), (5,12,13), (8,15,17)
• Sum of interior angles of n-gon: $(n-2) \times 180°$
• Volume of rectangular box: $l \times w \times h$

Pythagorean Theorem: a² + b² = c²

Example

A rectangle has perimeter 28 and width 5. Find the area.

Length = (28/2) − 5 = 9; Area = 9 × 5 = 45

④ Probability Probability

Core Rules:
• P(event) = favorable outcomes ÷ total outcomes
• P(A and B) = P(A) × P(B) [if independent]
• P(A or B) = P(A) + P(B) − P(A and B)
• P(not A) = 1 − P(A)

0 ≤ P(event) ≤ 1

Example

A bag has 3 red and 5 blue marbles. P(picking red)?

Answer: 3/(3+5) = 3/8

⑤ Statistics & Data Statistics

Key Definitions:
• Mean = sum ÷ count
• Median = middle value when sorted
• Mode = most frequent value
• Range = max − min

Mean = (x₁ + x₂ + ... + xₙ) / n

Example

Data set: {3, 7, 7, 9, 4}. Find median and mode.

Sorted: {3,4,7,7,9} → Median = 7, Mode = 7

⑥ Counting & Combinatorics Combinatorics

Counting Principles:
• Multiplication principle: if A has m ways and B has n ways → m × n ways
• Permutations: $P(n,r) = \frac{n!}{(n-r)!}$
• Combinations: $C(n,r) = \frac{n!}{r!(n-r)!}$
• Complementary counting: total − (what you DON'T want)

C(n,2) = n(n−1)/2

Example

How many ways to choose 2 students from 5?

C(5,2) = 5!/(2!·3!) = 10

AMC8

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