AMC 8 Master Practice | 20 Essential Problems

🔢 ① Number Theory

📌 Divisibility: A number is divisible by 3 if its digit sum is divisible by 3. By 9 if digit sum divisible by 9. By 4 if last two digits divisible by 4.

LCM(a,b) × GCD(a,b) = a × b

Number of factors of n = p₁^a · p₂^b … → (a+1)(b+1)…

📌 A prime number has exactly 2 factors: 1 and itself. 1 is NOT prime.

Example: How many positive factors does 36 have?

36 = 2² × 3² → (2+1)(2+1) = 9 factors: 1,2,3,4,6,9,12,18,36

➗ ② Arithmetic, Fractions & Percents

Percent change = (New − Old) / Old × 100%

Average = Sum of values / Number of values

📌 To add fractions: find common denominator. a/b + c/d = (ad+bc)/(bd)

📌 "X% of Y" = (X/100) × Y

Example: A shirt costs $40, then increases by 25%. New price?

$40 × 1.25 = $50

🔣 ③ Algebra & Patterns

Arithmetic sequence: aₙ = a₁ + (n−1)d

Sum of first n integers: n(n+1)/2

📌 To solve equations: perform same operation on both sides. Keep track of each step.

Example: Find the sum 1+2+3+…+100.

100 × 101 / 2 = 5050

📐 ④ Geometry

Area of rectangle = length × width

Area of triangle = (1/2) × base × height

Area of circle = π × r² | Circumference = 2πr

Pythagorean theorem: a² + b² = c²

📌 Sum of interior angles of a polygon with n sides = (n−2) × 180°

📌 Common Pythagorean triples: (3,4,5), (5,12,13), (8,15,17)

Example: Area of a right triangle with legs 6 and 8?

(1/2) × 6 × 8 = 24

🎲 ⑤ Counting & Probability

Multiplication principle: if A ways then B ways → A × B total ways

Combinations: C(n,r) = n! / (r!(n−r)!)

Probability = (Favorable outcomes) / (Total outcomes)

📌 P(A or B) = P(A) + P(B) – P(A and B) for non-mutually exclusive events

Example: How many ways to choose 2 students from 5?

C(5,2) = 5!/(2!×3!) = 10 ways

🧩 ⑥ Logic & Word Problems

📌 Rate × Time = Distance (or Work). For combined rates: add individual rates.

If A does job in a hours, rate = 1/a job per hour

📌 Draw a diagram or table for multi-step word problems. Label every variable.

Example: A tap fills a tank in 4 hours, another in 6 hours. Together?

Rate = 1/4 + 1/6 = 5/12. Time = 12/5 = 2.4 hours

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Core Concepts & Key Formulas

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Core Concepts & Key Formulas

Master
Practice
Workbook