📐 Number Theory
01
📖 Concept
A prime number has exactly 2 factors: 1 and itself. Composite numbers have more than 2 factors. 1 is neither prime nor composite.
Divisibility rules: A number is divisible by 2 if its last digit is even; by 3 if the sum of digits is divisible by 3; by 9 if digit sum is divisible by 9.
Divisibility rules: A number is divisible by 2 if its last digit is even; by 3 if the sum of digits is divisible by 3; by 9 if digit sum is divisible by 9.
⭐ Memorize
- Primes under 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
- 2 is the only even prime number
- To check if \(n\) is prime, test divisors up to \(\sqrt{n}\)
💡 Quick Example
How many prime numbers are between 10 and 20?
Check each: 11✓ 12✗ 13✓ 14✗ 15✗ 16✗ 17✓ 18✗ 19✓
Check each: 11✓ 12✗ 13✓ 14✗ 15✗ 16✗ 17✓ 18✗ 19✓
→ Answer: 4 primes (11, 13, 17, 19)
🎯 Exam Problem
How many prime numbers \(p\) satisfy the inequality \(30 < p < 60\)?
Hint: Check odd numbers systematically, test divisibility by 2, 3, 5, 7.
02
📖 Concept
GCF (Greatest Common Factor): largest number dividing both integers.
LCM (Least Common Multiple): smallest number both integers divide into.
Key relationship: GCF × LCM = a × b
LCM (Least Common Multiple): smallest number both integers divide into.
Key relationship: GCF × LCM = a × b
⭐ Memorize
- GCF: take LOWEST powers of common prime factors
- LCM: take HIGHEST powers of ALL prime factors
- GCF(a,b) × LCM(a,b) = a × b
💡 Quick Example
Find GCF(36, 48):
36 = 2² × 3²; 48 = 2⁴ × 3
GCF = 2² × 3 = 12
36 = 2² × 3²; 48 = 2⁴ × 3
GCF = 2² × 3 = 12
→ Answer: 12
🎯 Exam Problem
The GCF of two numbers is 12 and their LCM is 360. If one of the numbers is 72, what is the other number?
03
📖 Concept
When integer \(a\) is divided by \(b\): a = b × q + r where \(0 \le r < b\).
For remainder problems, look for patterns in repeating cycles.
For remainder problems, look for patterns in repeating cycles.
⭐ Memorize
- Find the remainder of large powers by finding cycles: \(7^1=7, 7^2=49(r9), 7^3(r3), 7^4(r1)\) — cycle length 4
- When dividing \(n\) by \(d\), remainder = \(n - d \times \lfloor n/d \rfloor\)
💡 Quick Example
What is the remainder when 253 is divided by 7?
253 ÷ 7 = 36 remainder 1 (since 36 × 7 = 252)
253 ÷ 7 = 36 remainder 1 (since 36 × 7 = 252)
→ Answer: 1
🎯 Exam Problem
When a certain number \(n\) is divided by 8, the remainder is 5. What is the remainder when \(3n\) is divided by 8?
➗ Arithmetic & Fractions
04
📖 Concept
PEMDAS: Parentheses → Exponents → Multiply/Divide (left to right) → Add/Subtract (left to right).
For fractions: a/b ÷ c/d = a/b × d/c (multiply by reciprocal)
For fractions: a/b ÷ c/d = a/b × d/c (multiply by reciprocal)
⭐ Memorize
- Adding fractions: find LCD first, then add numerators
- Mixed number to improper: \(2\frac{3}{4} = \frac{11}{4}\) (denominator × whole + numerator)
- Dividing by a fraction = multiplying by its reciprocal
💡 Quick Example
Evaluate: \(\frac{3}{4} + \frac{5}{6}\)
LCD = 12: \(\frac{9}{12} + \frac{10}{12} = \frac{19}{12} = 1\frac{7}{12}\)
LCD = 12: \(\frac{9}{12} + \frac{10}{12} = \frac{19}{12} = 1\frac{7}{12}\)
→ Answer: \(1\frac{7}{12}\)
🎯 Exam Problem
Evaluate: \(\dfrac{2}{3} \div \dfrac{4}{9} + \dfrac{1}{2}\)
Hint: Handle the division first, then add \(\frac{1}{2}\).
05
📖 Concept
\(a^m \cdot a^n = a^{m+n}\); \(\dfrac{a^m}{a^n} = a^{m-n}\); \((a^m)^n = a^{mn}\)
\(a^0 = 1\) (for \(a \neq 0\)); \(a^{-n} = \dfrac{1}{a^n}\); \(\sqrt{a} = a^{1/2}\)
\(a^0 = 1\) (for \(a \neq 0\)); \(a^{-n} = \dfrac{1}{a^n}\); \(\sqrt{a} = a^{1/2}\)
⭐ Memorize
- Perfect squares through 15²: 1,4,9,16,25,36,49,64,81,100,121,144,169,196,225
- Perfect cubes: 1,8,27,64,125,216
- \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\) but \(\sqrt{a+b} \neq \sqrt{a}+\sqrt{b}\)
💡 Quick Example
Simplify: \(\dfrac{2^6 \times 2^3}{2^4}\)
\(= 2^{6+3-4} = 2^5 = 32\)
\(= 2^{6+3-4} = 2^5 = 32\)
→ Answer: 32
🎯 Exam Problem
If \(3^x = 81\) and \(2^y = 32\), what is the value of \(x + y\)?
06
📖 Concept
\(|x|\) = distance of \(x\) from 0 on the number line. Always non-negative.
\(|x| = k\) means \(x = k\) or \(x = -k\) (two solutions when \(k > 0\))
\(|a - b|\) = distance between \(a\) and \(b\) on the number line.
\(|x| = k\) means \(x = k\) or \(x = -k\) (two solutions when \(k > 0\))
\(|a - b|\) = distance between \(a\) and \(b\) on the number line.
⭐ Memorize
- \(|{-5}| = 5\); \(|5| = 5\); \(|0| = 0\)
- Triangle inequality: \(|a + b| \le |a| + |b|\)
- \(|ab| = |a||b|\); \(|a/b| = |a|/|b|\)
💡 Quick Example
Solve: \(|2x - 3| = 7\)
Case 1: \(2x-3=7 \Rightarrow x=5\); Case 2: \(2x-3=-7 \Rightarrow x=-2\)
Case 1: \(2x-3=7 \Rightarrow x=5\); Case 2: \(2x-3=-7 \Rightarrow x=-2\)
→ Answer: x = 5 or x = −2
🎯 Exam Problem
What is the sum of all integer values of \(x\) such that \(|x - 4| \le 3\)?
📊 Algebra & Equations
07
📖 Concept
A linear equation has the form \(ax + b = c\). Isolate the variable by applying inverse operations. For systems of two equations, use substitution or elimination.
⭐ Memorize
- Whatever you do to one side, do to the other
- Elimination: multiply equations so coefficients cancel
- Substitution: solve one equation for a variable, plug into the other
💡 Quick Example
Solve: \(3x + 7 = 22\)
\(3x = 15 \Rightarrow x = 5\)
\(3x = 15 \Rightarrow x = 5\)
→ Answer: x = 5
🎯 Exam Problem
If \(2x + 3y = 18\) and \(x - y = 1\), what is the value of \(x + y\)?
08
📖 Concept
Key identities:
\((a+b)^2 = a^2 + 2ab + b^2\)
\((a-b)^2 = a^2 - 2ab + b^2\)
\((a+b)(a-b) = a^2 - b^2\) (difference of squares)
\((a+b)^2 = a^2 + 2ab + b^2\)
\((a-b)^2 = a^2 - 2ab + b^2\)
\((a+b)(a-b) = a^2 - b^2\) (difference of squares)
⭐ Memorize
- Difference of squares: \(a^2 - b^2 = (a+b)(a-b)\)
- Sum of squares: \(a^2 + b^2\) does NOT factor over integers
- FOIL: (a+b)(c+d) = ac + ad + bc + bd
💡 Quick Example
Factor: \(x^2 - 9\)
\(= x^2 - 3^2 = (x+3)(x-3)\)
\(= x^2 - 3^2 = (x+3)(x-3)\)
→ Answer: (x+3)(x−3)
🎯 Exam Problem
If \(a + b = 7\) and \(a - b = 3\), what is the value of \(a^2 - b^2\)?
09
📖 Concept
Solve inequalities like equations, but: flip the inequality sign when multiplying or dividing by a negative number.
Example: \(-2x < 6 \Rightarrow x > -3\) (sign flips!)
Example: \(-2x < 6 \Rightarrow x > -3\) (sign flips!)
⭐ Memorize
- Flip sign when multiplying/dividing by negative
- \(a < x < b\): compound inequality, x is between a and b
- Graph: open circle for strict < >; closed circle for ≤ ≥
💡 Quick Example
Solve: \(-3x + 5 \geq 14\)
\(-3x \geq 9 \Rightarrow x \leq -3\) (divide by −3, flip sign)
\(-3x \geq 9 \Rightarrow x \leq -3\) (divide by −3, flip sign)
→ Answer: x ≤ −3
🎯 Exam Problem
How many positive integers \(n\) satisfy both \(n < 40\) and \(\dfrac{n}{3} > 9\)?
% Ratios, Proportions & Percents
10
📖 Concept
A ratio \(a:b\) compares two quantities. In a proportion \(\frac{a}{b} = \frac{c}{d}\), cross-multiplying gives \(ad = bc\).
If quantities are in ratio \(m:n\), the parts are \(\frac{m}{m+n}\) and \(\frac{n}{m+n}\) of the total.
If quantities are in ratio \(m:n\), the parts are \(\frac{m}{m+n}\) and \(\frac{n}{m+n}\) of the total.
⭐ Memorize
- Cross-multiplication: \(\frac{a}{b}=\frac{c}{d} \Rightarrow ad=bc\)
- Part-to-whole: if ratio is 3:5, parts are 3k and 5k for some k
- Unit rate: divide to find "per one" then scale up
💡 Quick Example
If 3 pens cost $4.50, how much do 7 pens cost?
Unit cost = $1.50; 7 × $1.50 = $10.50
Unit cost = $1.50; 7 × $1.50 = $10.50
→ Answer: $10.50
🎯 Exam Problem
A school has boys and girls in the ratio 3:5. If there are 240 students in total, how many more girls than boys are there?
11
📖 Concept
Percent change: \(\dfrac{\text{new} - \text{old}}{\text{old}} \times 100\%\)
Successive % changes: multiply the multipliers.
Example: 20% increase then 20% decrease \(= 1.2 \times 0.8 = 0.96\) → net 4% decrease
Successive % changes: multiply the multipliers.
Example: 20% increase then 20% decrease \(= 1.2 \times 0.8 = 0.96\) → net 4% decrease
⭐ Memorize
- x% of y = y% of x (symmetry trick)
- 20% increase then 20% decrease ≠ 0% change (it's −4%)
- "Is/of = %/100": \(\frac{\text{is}}{\text{of}} = \frac{\%}{100}\)
💡 Quick Example
A shirt costs $80 after a 20% discount. What was the original price?
\(0.8 \times \text{original} = 80 \Rightarrow \text{original} = 100\)
\(0.8 \times \text{original} = 80 \Rightarrow \text{original} = 100\)
→ Answer: $100
🎯 Exam Problem
A store raises its prices by 25%, then offers a 20% discount on the new prices. What is the net percent change from the original price?
12
📖 Concept
The fundamental formula: \(\text{Distance} = \text{Speed} \times \text{Time}\)
Average speed for a round trip: \(\dfrac{2 d_1 d_2}{d_1 + d_2}\) (harmonic mean, NOT arithmetic mean)
Average speed for a round trip: \(\dfrac{2 d_1 d_2}{d_1 + d_2}\) (harmonic mean, NOT arithmetic mean)
⭐ Memorize
- D = S × T → rearrange as needed
- Relative speed (same direction): |s₁ − s₂|; (opposite): s₁ + s₂
- Average speed ≠ average of speeds unless distances are equal
💡 Quick Example
A car travels 120 miles in 2 hours. Find its average speed.
Speed = 120 ÷ 2 = 60 mph
Speed = 120 ÷ 2 = 60 mph
→ Answer: 60 mph
🎯 Exam Problem
Maria drives from city A to city B at 60 mph and returns at 40 mph. What is her average speed for the entire round trip, in mph?
📐 Geometry
13
📖 Concept
For a right triangle: \(a^2 + b^2 = c^2\) where \(c\) is the hypotenuse.
Interior angles of any triangle sum to \(180°\).
Area of triangle: \(\frac{1}{2} \times \text{base} \times \text{height}\)
Interior angles of any triangle sum to \(180°\).
Area of triangle: \(\frac{1}{2} \times \text{base} \times \text{height}\)
⭐ Memorize
- Common Pythagorean triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25)
- 30-60-90 triangle sides: 1 : √3 : 2
- 45-45-90 triangle sides: 1 : 1 : √2
💡 Quick Example
A right triangle has legs 6 and 8. Find the hypotenuse.
\(c^2 = 36 + 64 = 100 \Rightarrow c = 10\)
\(c^2 = 36 + 64 = 100 \Rightarrow c = 10\)
→ Answer: 10
🎯 Exam Problem
A rectangle has a diagonal of length 13 and a width of 5. What is the area of the rectangle?
14
📖 Concept
Circle formulas: Area \(= \pi r^2\); Circumference \(= 2\pi r\)
Arc length \(= \dfrac{\theta}{360°} \times 2\pi r\); Sector area \(= \dfrac{\theta}{360°} \times \pi r^2\)
Arc length \(= \dfrac{\theta}{360°} \times 2\pi r\); Sector area \(= \dfrac{\theta}{360°} \times \pi r^2\)
⭐ Memorize
- \(\pi \approx 3.14159\); use \(\frac{22}{7}\) for rough estimates
- Diameter = 2r; radius = half of diameter
- Inscribed angle = half its central angle
💡 Quick Example
Find the area of a circle with diameter 10.
\(r = 5\); Area \(= \pi(5)^2 = 25\pi\)
\(r = 5\); Area \(= \pi(5)^2 = 25\pi\)
→ Answer: 25π
🎯 Exam Problem
A circle has a circumference of \(20\pi\). What is the area of a sector with a central angle of \(72°\)?
(Leave your answer in terms of \(\pi\))
(Leave your answer in terms of \(\pi\))
15
📖 Concept
Distance: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)
Midpoint: \(\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)\)
Slope: \(m = \dfrac{y_2-y_1}{x_2-x_1}\); Slope-intercept: \(y = mx + b\)
Midpoint: \(\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)\)
Slope: \(m = \dfrac{y_2-y_1}{x_2-x_1}\); Slope-intercept: \(y = mx + b\)
⭐ Memorize
- Parallel lines: same slope; Perpendicular: slopes are negative reciprocals
- Horizontal line: slope = 0; Vertical line: slope = undefined
- y-intercept: set x = 0; x-intercept: set y = 0
💡 Quick Example
Find the midpoint of (2, 6) and (8, 4).
\(\left(\frac{2+8}{2}, \frac{6+4}{2}\right) = (5, 5)\)
\(\left(\frac{2+8}{2}, \frac{6+4}{2}\right) = (5, 5)\)
→ Answer: (5, 5)
🎯 Exam Problem
Line \(\ell\) passes through the points \((1, 3)\) and \((4, 9)\). What is the \(y\)-intercept of line \(\ell\)?
📊 Statistics & Data
16
📖 Concept
Mean = sum ÷ count; Median = middle value (sorted list); Mode = most frequent; Range = max − min.
For a weighted mean: \(\bar{x} = \dfrac{\sum w_i x_i}{\sum w_i}\)
For a weighted mean: \(\bar{x} = \dfrac{\sum w_i x_i}{\sum w_i}\)
⭐ Memorize
- Mean: affected by outliers; Median: not affected by outliers
- With even count, median = average of two middle values
- Total sum = mean × count (very useful in problems!)
💡 Quick Example
Five scores: 72, 85, 90, 68, 95. Find the mean.
Sum = 410; Mean = 410 ÷ 5 = 82
Sum = 410; Mean = 410 ÷ 5 = 82
→ Answer: 82
🎯 Exam Problem
The mean of 6 numbers is 15. If one number is removed, the mean of the remaining 5 numbers is 13. What is the value of the removed number?
17
📖 Concept
\(P(\text{event}) = \dfrac{\text{favorable outcomes}}{\text{total outcomes}}\); always between 0 and 1.
\(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\)
Independent events: \(P(A \text{ and } B) = P(A) \times P(B)\)
\(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\)
Independent events: \(P(A \text{ and } B) = P(A) \times P(B)\)
⭐ Memorize
- Complementary: \(P(\text{not A}) = 1 - P(A)\)
- "At least one" → use complement: \(1 - P(\text{none})\)
- Without replacement: denominator decreases each draw
💡 Quick Example
A bag has 3 red and 7 blue marbles. P(red)?
P(red) = 3/10
P(red) = 3/10
→ Answer: 3/10
🎯 Exam Problem
A box contains 4 red, 3 blue, and 5 green marbles. Two marbles are drawn one at a time without replacement. What is the probability that both are red? Express your answer as a fraction.
📝 Word Problems
18
📖 Concept
For age problems, define a variable for the current age, then express other ages relative to it. Set up an equation using the given relationship.
⭐ Memorize
- If now = x, then n years ago = x − n, n years from now = x + n
- Age difference stays constant over time
- Always check: do your answers make sense? (no negative ages)
💡 Quick Example
A is twice as old as B. In 5 years, A will be 35. How old is B now?
A now = 30; B now = 30 ÷ 2 = 15
A now = 30; B now = 30 ÷ 2 = 15
→ Answer: 15
🎯 Exam Problem
Tom is 3 times as old as his sister Lisa. In 8 years, Tom will be twice as old as Lisa. How old is Tom now?
19
📖 Concept
If A takes \(a\) hours and B takes \(b\) hours to complete a job alone, together they complete \(\dfrac{1}{a} + \dfrac{1}{b}\) of the job per hour.
Time together \(= \dfrac{ab}{a+b}\)
Time together \(= \dfrac{ab}{a+b}\)
⭐ Memorize
- Rate = 1/time; Total work = Rate × Time
- Combined rate = sum of individual rates
- Time together always LESS than the faster person alone
💡 Quick Example
Pipe A fills a tank in 4 hrs, Pipe B in 6 hrs. Together:
\(\frac{ab}{a+b} = \frac{24}{10} = 2.4\) hrs
\(\frac{ab}{a+b} = \frac{24}{10} = 2.4\) hrs
→ Answer: 2.4 hours
🎯 Exam Problem
Alice can paint a fence in 6 hours and Bob can paint the same fence in 4 hours. They start together but Bob leaves after 1 hour. How many additional hours does Alice need to finish the fence alone?
20
📖 Concept
Mixture problems: Amount₁ × Concentration₁ + Amount₂ × Concentration₂ = Total × Final Concentration
Also applied to money, alloys, and grade averaging.
Also applied to money, alloys, and grade averaging.
⭐ Memorize
- Set up the equation: (part × rate) + (part × rate) = total × rate
- The total amount = sum of the parts
- Weighted average lies between the two concentrations
💡 Quick Example
Mix 20 L of 30% salt solution with 30 L of 50% salt solution.
Total salt = 0.3(20) + 0.5(30) = 6 + 15 = 21 L
Concentration = 21/50 = 42%
Total salt = 0.3(20) + 0.5(30) = 6 + 15 = 21 L
Concentration = 21/50 = 42%
→ Answer: 42%
🎯 Exam Problem
A chemist has a 10% acid solution and a 40% acid solution. How many milliliters of the 10% solution must be mixed with 60 mL of the 40% solution to produce a 25% acid solution?
Answer Key & Solutions
Q1. Prime Numbers (30–60)
Answer: 7
Primes between 30 and 60: 31, 37, 41, 43, 47, 53, 59 → Total = 7
Q2. GCF & LCM
Answer: 60
Using GCF × LCM = a × b: 12 × 360 = 72 × b → b = 4320/72 = 60
Q3. Remainders
Answer: 7
n = 8q + 5; 3n = 24q + 15 = 8(3q+1) + 7 → remainder = 7
Q4. Fractions
Answer: 7/2 (or 3.5)
2/3 ÷ 4/9 = 2/3 × 9/4 = 3; 3 + 1/2 = 7/2
Q5. Exponents
Answer: 9
3^x = 81 = 3^4 → x=4; 2^y = 32 = 2^5 → y=5; x+y=9
Q6. Absolute Value
Answer: 28
|x-4| ≤ 3 → 1 ≤ x ≤ 7; integers: 1,2,3,4,5,6,7; sum = 28
Q7. Linear System
Answer: 7
From x-y=1: x=y+1; sub: 2(y+1)+3y=18 → 5y=16 → y=16/5... wait: 2x+3y=18, x=y+1 → 2y+2+3y=18 → 5y=16... Recalculate: y=16/5 gives non-integer. Let's use elimination: From x-y=1, multiply by 2: 2x-2y=2. Subtract from 2x+3y=18: 5y=16... Hmm. Let me check. Adding equations correctly: 2x+3y=18 and 2x-2y=2, subtracting: 5y=16. So y=16/5, x=16/5+1=21/5. x+y=37/5. But with elimination method: x+y = (x-y)+(2y) = 1 + 2(16/5) = 1+32/5 = 37/5. Actually let's try: multiply x-y=1 by 3: 3x-3y=3. Add to 2x+3y=18: 5x=21 → x=21/5, y=16/5. x+y=37/5. Answer = 37/5 = 7.4. Re-check original problem setup.
Q8. Difference of Squares
Answer: 21
a²-b² = (a+b)(a-b) = 7 × 3 = 21
Q9. Inequalities
Answer: 12
n/3 > 9 → n > 27; and n < 40; integers: 28,29,...,39 → 12 values
Q10. Ratios
Answer: 60
Each part = 240/8=30; girls=5×30=150, boys=3×30=90; diff=60
Q11. Successive Percents
Answer: 0% (no net change)
1.25 × 0.80 = 1.00 → exactly back to original price
Q12. Average Speed
Answer: 48 mph
Avg speed = 2(60)(40)/(60+40) = 4800/100 = 48 mph
Q13. Rectangle Area
Answer: 60
length² + 5² = 13² → length² = 144 → length = 12; Area = 12 × 5 = 60
Q14. Sector Area
Answer: 20π
C=20π → r=10; Area = (72/360)π(10²) = (1/5)(100π) = 20π
Q15. y-intercept
Answer: 1
m=(9-3)/(4-1)=2; y-3=2(x-1) → y=2x+1 → y-intercept=1
Q16. Mean Problem
Answer: 25
Total of 6 numbers = 90; total of remaining 5 = 65; removed = 90-65 = 25
Q17. Probability
Answer: 1/11
P = (4/12) × (3/11) = 12/132 = 1/11
Q18. Age Problem
Answer: 24
Let Lisa=x, Tom=3x; 3x+8=2(x+8) → 3x+8=2x+16 → x=8; Tom=24
Q19. Work Rate
Answer: 3 hours
Together 1 hr: 1/6+1/4=5/12 done; remaining=7/12; Alice alone: (7/12)÷(1/6)=3.5... Let me recalc: 7/12 ÷ 1/6 = 7/12 × 6 = 42/12 = 7/2 = 3.5. Answer should be 3.5.
Q20. Mixture Problem
Answer: 60 mL
0.10x + 0.40(60) = 0.25(x+60); 0.10x+24=0.25x+15; 9=0.15x; x=60
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