AMC 8 Master Workbook

🔢

Number Theory

Divisibility, GCD, LCM, Primes

LCM(a,b) × GCD(a,b) = a × b Prime factorization: find all prime divisors # of factors of n = (e₁+1)(e₂+1)··· where n = p₁^e₁ · p₂^e₂ ···

A number is divisible by 3 if the sum of its digits is divisible by 3
A number is divisible by 9 if the sum of its digits is divisible by 9
A number is divisible by 4 if its last two digits form a number divisible by 4
To find GCD: use prime factorization, take the lowest powers
To find LCM: use prime factorization, take the highest powers

Example

How many positive factors does 60 have?

60 = 2² × 3¹ × 5¹ → (2+1)(1+1)(1+1) = 12 factors

➗

Fractions, Ratios & Percentages

Essential arithmetic relationships

% change = (new − old) / old × 100% If a:b = m:n then a = mk, b = nk for some k To compare fractions: cross-multiply

To add fractions: find common denominator first
Percent of a number: multiply by decimal (e.g. 30% of 80 = 0.30 × 80 = 24)
"A is 20% more than B" means A = 1.2B
If ratio is a:b, the parts are a/(a+b) and b/(a+b) of the whole

Example

A price decreases from $50 to $40. What is the percent decrease?

(50 − 40)/50 × 100% = 10/50 × 100% = 20%

📐

Geometry

Area, Perimeter, Angles, Pythagorean Theorem

Area: Rectangle = lw, Triangle = ½bh, Circle = πr² Perimeter: Rectangle = 2(l+w), Circle = 2πr Pythagorean Theorem: a² + b² = c² Sum of interior angles of n-gon = (n−2)×180°

Common right triangles: 3-4-5, 5-12-13, 8-15-17 (and their multiples)
Area of trapezoid = ½(b₁ + b₂) × h
Vertical angles are equal; supplementary angles add to 180°
In a rectangle, diagonal² = length² + width²

Example

A right triangle has legs 5 and 12. Find the hypotenuse.

c² = 5² + 12² = 25 + 144 = 169, so c = 13

🔤

Algebra & Equations

Linear equations, word problems, sequences

Arithmetic sequence: aₙ = a₁ + (n−1)d Sum of arithmetic seq: S = n(a₁ + aₙ)/2 Distance = Rate × Time Work: 1/T = 1/T₁ + 1/T₂

Set up equations by letting a variable represent the unknown
For consecutive integers starting at n: n, n+1, n+2, ...
Arithmetic mean = sum ÷ count
If total changes, recompute the average carefully

Example

The 1st term of an arithmetic sequence is 3 and the common difference is 4. What is the 8th term?

a₈ = 3 + (8−1)×4 = 3 + 28 = 31

🎲

Probability & Combinatorics

Counting, combinations, basic probability

P(event) = favorable outcomes / total outcomes Combination: C(n,r) = n! / (r!(n−r)!) Permutation: P(n,r) = n! / (n−r)! Multiplication Principle: m × n ways for 2 independent events

P(A and B) = P(A) × P(B) if independent
P(A or B) = P(A) + P(B) − P(A and B)
Complement: P(not A) = 1 − P(A)
Arrangements of n distinct items: n! ways

Example

How many ways can you choose 2 students from a class of 5?

C(5,2) = 5!/(2!·3!) = (5×4)/2 = 10 ways

📊

Statistics & Data

Mean, Median, Mode, Range

Mean = (sum of all values) ÷ (count of values) Median = middle value when sorted Mode = most frequently occurring value Range = maximum − minimum

When data is sorted, median is the middle value (or average of two middle values for even count)
Adding a constant to all values shifts mean and median by that constant
Outliers affect mean significantly but not median

Example

Find the median of: 3, 7, 2, 9, 5

Sorted: 2, 3, 5, 7, 9 → Median = 5

🔄

Patterns & Logical Reasoning

Number patterns, digit problems, logic

Sum 1 to n = n(n+1)/2 Sum of odd numbers 1 to (2n−1) = n² Look for cycles in repeating patterns

For digit problems: a 2-digit number with digits a,b equals 10a + b
In repeating patterns, find the cycle length then use remainder
The units digit of powers of 2 cycles: 2, 4, 8, 6, 2, 4, 8, 6, ...
Always verify by testing small cases first

Example

What is the units digit of 2^(2023)?

Cycle of units digits of powers of 2: 2,4,8,6 (period 4). 2023 mod 4 = 3. Third in cycle = 8

Select your answer. Instant feedback provided.

Problem01

Number Theory

What is the greatest common divisor (GCD) of 48 and 72?

✦ Step-by-Step Solution

①Factor 48: 48 = 2⁴ × 3¹

②Factor 72: 72 = 2³ × 3²

③GCD = product of lowest powers: 2³ × 3¹ = 8 × 3 = 24

Answer: C (24)

Problem02

Number Theory

How many positive integer factors does 120 have?

✦ Step-by-Step Solution

①Prime factorize: 120 = 2³ × 3¹ × 5¹

②Number of factors: (3+1)(1+1)(1+1) = 4 × 2 × 2 = 16

Answer: D (16)

Problem03

Geometry

A rectangle has a length of 12 cm and a width of 5 cm. What is the length of its diagonal?

✦ Step-by-Step Solution

①The diagonal of a rectangle is: d = √(l² + w²)

②d = √(12² + 5²) = √(144 + 25) = √169 = 13

③This is the famous 5-12-13 Pythagorean triple!

Answer: C (13 cm)

Problem04

Geometry

A triangle has vertices at (0,0), (6,0), and (0,8). What is its area?

✦ Step-by-Step Solution

①This is a right triangle with legs along the axes: base = 6, height = 8

②Area = ½ × base × height = ½ × 6 × 8 = 24

Answer: C (24)

Problem05

Algebra

If 3x − 7 = 14, what is the value of x?

✦ Step-by-Step Solution

①Add 7 to both sides: 3x = 14 + 7 = 21

②Divide both sides by 3: x = 21 / 3 = 7

③Verify: 3(7) − 7 = 21 − 7 = 14 ✓

Answer: C (7)

Problem06

Algebra

The sum of three consecutive odd integers is 57. What is the largest of the three integers?

✦ Step-by-Step Solution

①Let the three consecutive odd integers be: n, n+2, n+4

②n + (n+2) + (n+4) = 57 → 3n + 6 = 57 → 3n = 51 → n = 17

③The three integers are 17, 19, 21. The largest is n+4 = 21

Answer: C (21)

Problem07

Probability

A bag contains 4 red, 3 blue, and 5 green marbles. One marble is drawn at random. What is the probability it is NOT green?

✦ Step-by-Step Solution

①Total marbles: 4 + 3 + 5 = 12

②P(green) = 5/12

③P(NOT green) = 1 − 5/12 = 7/12

Answer: B (7/12)

Problem08

Probability

Two fair six-sided dice are rolled. What is the probability that the sum equals 8?

✦ Step-by-Step Solution

①Total outcomes: 6 × 6 = 36

②Pairs that sum to 8: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 pairs

③Probability = 5/36

Answer: B (5/36)

Problem09

Statistics

The scores of 5 students are: 72, 85, 90, 68, and 95. What is the mean score?

✦ Step-by-Step Solution

①Sum: 72 + 85 + 90 + 68 + 95 = 410

②Mean: 410 / 5 = 82

Answer: B or C (82)

Problem10

Statistics

The average of 6 numbers is 14. When a 7th number is added, the average becomes 15. What is the 7th number?

✦ Step-by-Step Solution

①Sum of original 6 numbers: 6 × 14 = 84

②Sum of all 7 numbers: 7 × 15 = 105

③7th number: 105 − 84 = 21

Answer: D (21)

Problem11

Combinatorics

In how many ways can 4 students be seated in a row of 4 chairs?

✦ Step-by-Step Solution

①This is a permutation: arrange 4 distinct people in 4 seats

②Choices: 4 × 3 × 2 × 1 = 4! = 24

Answer: D (24)

Problem12

Combinatorics

A committee of 3 people is chosen from a group of 7. How many different committees are possible?

✦ Step-by-Step Solution

①Order doesn't matter for a committee, so use combinations

②C(7,3) = 7! / (3! × 4!) = (7 × 6 × 5) / (3 × 2 × 1) = 210/6 = 35

Answer: C (35)

Problem13

Patterns

What is the sum of all integers from 1 to 50?

✦ Step-by-Step Solution

①Use the formula: S = n(n+1)/2

②S = 50 × 51 / 2 = 2550 / 2 = 1275

Answer: B (1275)

Problem14

Number Theory

What is the units digit of 7^(2024)?

✦ Step-by-Step Solution

①Units digits of powers of 7 cycle with period 4: 7¹→7, 7²→9, 7³→3, 7⁴→1, then repeats

②2024 ÷ 4 = 506 remainder 0

③Remainder 0 corresponds to the 4th position in the cycle: units digit = 1... wait, let's recheck: cycle is 7,9,3,1. Remainder 0 → position 4 → units digit = 1

④Actually: 7⁴ = 2401, units digit 1. 7^(2024) = (7⁴)^(506), units digit = 1^(506) = 1. So the answer is 1.

Answer: A (1)

Problem15

Geometry

The sum of interior angles of a regular polygon is 1080°. How many sides does it have?

✦ Step-by-Step Solution

①Formula: sum of angles = (n − 2) × 180°

②Set equal to 1080: (n − 2) × 180 = 1080

③n − 2 = 1080 / 180 = 6 → n = 8

Answer: C (8 sides — octagon)

Problem16

Algebra

A train travels at 60 mph for 2.5 hours, then at 80 mph for 1.5 hours. What is the total distance traveled?

✦ Step-by-Step Solution

①First segment: 60 × 2.5 = 150 miles

②Second segment: 80 × 1.5 = 120 miles

③Total: 150 + 120 = 270 miles

Answer: C (270 miles)

Problem17

Number Theory

What is the LCM of 12, 18, and 24?

✦ Step-by-Step Solution

①12 = 2² × 3¹

②18 = 2¹ × 3²

③24 = 2³ × 3¹

④LCM = highest powers: 2³ × 3² = 8 × 9 = 72

Answer: D (72)

Problem18

Algebra

A store marks up the price of an item by 40%, then offers a 20% discount. What is the overall percent change from the original price?

✦ Step-by-Step Solution

①Let original price = P

②After 40% markup: P × 1.40 = 1.4P

③After 20% discount: 1.4P × 0.80 = 1.12P

④Net change: 1.12P − P = 0.12P, which is a 12% increase

Answer: B (12% increase)

Problem19

Geometry

A circle is inscribed in a square with side length 10. What is the area of the region inside the square but outside the circle? (Use π ≈ 3.14)

✦ Step-by-Step Solution

①Area of square: 10² = 100

②Inscribed circle has radius = half the side = 5

③Area of circle: π × 5² = 3.14 × 25 = 78.5

④Remaining area: 100 − 78.5 = 21.5

Answer: B (21.5)

Problem20

Algebra

The 5th term of an arithmetic sequence is 23 and the 9th term is 43. What is the first term of the sequence?

✦ Step-by-Step Solution

①From term 5 to term 9 there are 4 steps: 43 − 23 = 20, so common difference d = 20/4 = 5

②Use term 5: a₁ + 4d = 23 → a₁ + 4(5) = 23 → a₁ = 23 − 20 = 3

③Verify: 3, 8, 13, 18, 23 ✓ (5th term is 23)

Answer: C (3)

Final Score

0/20

0% correct

Complete Solutions

Detailed step-by-step explanations for all 20 problems

GCD of 48 and 72 — Number Theory

✓ Answer: C — 24

48 = 2⁴ × 3¹ | 72 = 2³ × 3² → GCD = 2³ × 3¹ = 8 × 3 = 24

Take the prime factorization of each number. For GCD, take the lowest power of each prime that appears in both factorizations.

Number of factors of 120 — Number Theory

✓ Answer: D — 16

120 = 2³ × 3¹ × 5¹ → (3+1)(1+1)(1+1) = 4 × 2 × 2 = 16

To count factors, add 1 to each exponent in the prime factorization, then multiply those results together.

Diagonal of 12×5 rectangle — Geometry

✓ Answer: C — 13 cm

d = √(12² + 5²) = √(144 + 25) = √169 = 13

The diagonal of a rectangle forms a right triangle with the sides. Apply the Pythagorean theorem. This is the 5-12-13 Pythagorean triple.

Area of triangle at (0,0),(6,0),(0,8) — Geometry

✓ Answer: C — 24

Area = ½ × base × height = ½ × 6 × 8 = 24

The two legs of this right triangle lie along the x-axis (length 6) and y-axis (length 8). The right angle is at the origin.

Solve 3x − 7 = 14 — Algebra

✓ Answer: C — 7

3x = 14 + 7 = 21 → x = 7

Add 7 to both sides to isolate the term with x, then divide by 3. Always verify by substituting back: 3(7) − 7 = 14 ✓

Three consecutive odd integers sum to 57 — Algebra

✓ Answer: C — 21

n + (n+2) + (n+4) = 57 → 3n + 6 = 57 → n = 17

Consecutive odd integers differ by 2. Let them be n, n+2, n+4. Solve for n=17, so the three integers are 17, 19, 21. Largest = 21.

Probability NOT green marble — Probability

✓ Answer: B — 7/12

P(not green) = 1 − P(green) = 1 − 5/12 = 7/12

Total = 4+3+5 = 12 marbles. 5 are green. Use the complement rule: P(not A) = 1 − P(A).

Two dice sum to 8 — Probability

✓ Answer: B — 5/36

Pairs: (2,6),(3,5),(4,4),(5,3),(6,2) = 5 pairs out of 36

List all pairs that sum to 8 systematically. Total sample space is 6×6=36. There are exactly 5 favorable outcomes.

Mean of 5 test scores — Statistics

✓ Answer: 82

Mean = (72 + 85 + 90 + 68 + 95) / 5 = 410 / 5 = 82

Add all values carefully: 72+85=157, 157+90=247, 247+68=315, 315+95=410. Then divide by 5.

7th number when average rises — Statistics

✓ Answer: D — 21

6×14 = 84, 7×15 = 105, 7th number = 105 − 84 = 21

The key insight: sum = average × count. Find the total sum before and after adding the new number, then subtract to find it.

Seating 4 students in a row — Combinatorics

✓ Answer: D — 24

4! = 4 × 3 × 2 × 1 = 24

For the first seat: 4 choices. For the second: 3 remaining. Third: 2. Fourth: 1. Multiply: 4×3×2×1 = 24.

Committees of 3 from 7 people — Combinatorics

✓ Answer: C — 35

C(7,3) = 7! / (3! × 4!) = (7×6×5) / (3×2×1) = 210/6 = 35

Order doesn't matter in a committee, so use combinations. Compute C(7,3) by simplifying before multiplying.

Sum of 1 to 50 — Patterns

✓ Answer: B — 1275

S = n(n+1)/2 = 50×51/2 = 2550/2 = 1275

Use Gauss's formula. Pair up: (1+50)+(2+49)+...+(25+26) = 25 pairs each summing to 51 = 25×51 = 1275.

Units digit of 7^2024 — Patterns

✓ Answer: A — 1

Cycle of 7: 7¹→7, 7²→9, 7³→3, 7⁴→1 (period 4). 2024 ÷ 4 = 506 R 0 → position 4 → units digit 1

Powers of 7 have a units digit cycle of length 4: 7, 9, 3, 1. Since 2024 is divisible by 4 (remainder 0), the units digit is the 4th in the cycle: 1.

Polygon with angle sum 1080° — Geometry

✓ Answer: C — 8 sides

(n−2) × 180 = 1080 → n−2 = 6 → n = 8

The formula (n−2)×180° gives the sum of interior angles of any polygon with n sides. Solve for n.

Train distance problem — Algebra

✓ Answer: C — 270 miles

60 × 2.5 + 80 × 1.5 = 150 + 120 = 270

Distance = Rate × Time. Calculate each segment separately then add. Don't try to find an average speed — compute total distance directly.

LCM of 12, 18, 24 — Number Theory

✓ Answer: D — 72

12=2²·3, 18=2·3², 24=2³·3 → LCM = 2³ × 3² = 8 × 9 = 72

For LCM, take the HIGHEST power of each prime that appears in any of the factorizations. Verify: 72÷12=6 ✓, 72÷18=4 ✓, 72÷24=3 ✓

40% markup then 20% discount — Ratios

✓ Answer: B — 12% increase

P × 1.40 × 0.80 = 1.12P → 12% increase

Multiply the multipliers: 1.40 × 0.80 = 1.12. This means 12% above the original. Note: 40% up then 20% down does NOT equal 20% up — they compound multiplicatively.

Square minus inscribed circle — Geometry

✓ Answer: B — 21.5

100 − π(5²) = 100 − 3.14×25 = 100 − 78.5 = 21.5

The inscribed circle touches all four sides, so its diameter equals the side length (10), making radius = 5. Subtract the circle's area from the square's area.

Arithmetic sequence: find first term — Algebra

✓ Answer: C — 3

d = (43−23)/(9−5) = 20/4 = 5. a₁ = a₅ − 4d = 23 − 20 = 3

The difference between terms 9 and 5 spans 4 common differences. So d = 20/4 = 5. Work backward from term 5: subtract 4d to reach term 1.