01
Core Concepts
Number Theory
GCD, LCM & Factors
Use prime factorization to find GCD and LCM efficiently.
GCD × LCM = a × b
Factors of $n = p^a q^b \cdots$: $(a+1)(b+1)\cdots$
Factors of $n = p^a q^b \cdots$: $(a+1)(b+1)\cdots$
- Factor completely before comparing
- GCD = product of common primes (lowest power)
- LCM = product of all primes (highest power)
Fractions & Percents
Comparison & Change
Convert to decimals or common denominators to compare.
% change $= \dfrac{\text{new} - \text{old}}{\text{old}} \times 100$
- Successive discounts: multiply, don't add
- Cross-multiply to compare fractions quickly
- $9/15 = 3/5$ — always simplify first
Geometry
Area, Angles & Similar Figures
Know Pythagorean triples (3-4-5, 5-12-13, 8-15-17).
Triangle: $A = \tfrac{1}{2}bh$
Circle: $A = \pi r^2$, $C = 2\pi r$
Angle sum in $\triangle = 180°$
Circle: $A = \pi r^2$, $C = 2\pi r$
Angle sum in $\triangle = 180°$
- Similar figures: ratios scale linearly, areas square
- Rectangle: $P = 2(l+w)$, $A = lw$
Probability & Counting
Basic Probability & Combinations
List all outcomes systematically. Use complement when "not" appears.
$P(E) = \dfrac{\text{favorable}}{\text{total}}$
$C(n,r) = \dfrac{n!}{r!(n-r)!}$
$C(n,r) = \dfrac{n!}{r!(n-r)!}$
- $P(\text{not A}) = 1 - P(A)$
- Dice sum 7: pairs (1,6)(2,5)(3,4)(4,3)(5,2)(6,1) → 6/36
Algebra
Sequences & Word Problems
Arithmetic sequence: constant difference. Always define your variable clearly.
$a_n = a_1 + (n-1)d$
Rate problems: $d = rt$
Rate problems: $d = rt$
- Combined rate: add individual rates
- Two trains toward each other: combined speed
- Pool fill/drain: rates add or subtract
Statistics
Mean, Median, Mode, Ratio
Mean = sum ÷ count. Median = middle value after sorting. Mode = most frequent.
$\bar{x} = \dfrac{\sum x_i}{n}$
- Sort before finding median
- Ratio a:b → multiply by same $k$ to get actual amounts
02
Worked Examples
Example A · Number Theory
How many positive divisors does 36 have?
1
Prime factorize: $36 = 2^2 \times 3^2$
2
Number of divisors $= (2+1)(2+1) = 9$
3
Check: 1,2,3,4,6,9,12,18,36 — yes, 9 divisors ✓
Answer: 9
Example B · Probability
A bag has 2 red, 3 blue, 5 green marbles. What is P(not green)?
1
Total marbles = 2 + 3 + 5 = 10
2
Not green = 2 + 3 = 5
3
$P(\text{not green}) = \dfrac{5}{10} = \dfrac{1}{2}$
Answer: 1/2
Example C · Algebra (Sequence)
Find the 15th term of the sequence 2, 5, 8, 11, …
1
Identify: $a_1 = 2$, common difference $d = 3$
2
$a_{15} = 2 + (15-1)\times 3 = 2 + 42 = 44$
Answer: 44
Example D · Geometry
A right triangle has legs 5 and 12. Find its perimeter.
1
Hypotenuse: $\sqrt{5^2+12^2}=\sqrt{169}=13$
2
Perimeter $= 5 + 12 + 13 = 30$
Answer: 30
03
Practice Problems
01
Number Theory · LCM
★★★
What is the least common multiple (LCM) of 12 and 18?
📐 Solution
Prime factorize: $12 = 2^2 \times 3$ and $18 = 2 \times 3^2$.
LCM takes the highest power of each prime: $2^2 \times 3^2 = 4 \times 9 = \mathbf{36}$.
Verify: $36 \div 12 = 3$ ✓ and $36 \div 18 = 2$ ✓. The answer is C · 36.
LCM takes the highest power of each prime: $2^2 \times 3^2 = 4 \times 9 = \mathbf{36}$.
Verify: $36 \div 12 = 3$ ✓ and $36 \div 18 = 2$ ✓. The answer is C · 36.
02
Fractions · Comparison
★★★
Which of the following fractions has the greatest value?
📐 Solution
Convert all to decimals:
A: $3/5 = 0.60$ · B: $5/8 = 0.625$ · C: $7/10 = 0.70$ · D: $2/3 \approx 0.667$ · E: $9/15 = 3/5 = 0.60$
The greatest is $\mathbf{0.70} = 7/10$. The answer is C · 7/10.
A: $3/5 = 0.60$ · B: $5/8 = 0.625$ · C: $7/10 = 0.70$ · D: $2/3 \approx 0.667$ · E: $9/15 = 3/5 = 0.60$
The greatest is $\mathbf{0.70} = 7/10$. The answer is C · 7/10.
03
Percents · Successive Change
★★★
A shirt originally costs $40. It is discounted by 25%, then a 10% sales tax is added to the discounted price. What is the final price?
📐 Solution
After 25% discount: $\$40 \times 0.75 = \$30$.
After 10% tax: $\$30 \times 1.10 = \$33$.
Key warning: do NOT add $-25\%$ and $+10\%$ together. Multiply sequentially. The answer is C · $33.00.
After 10% tax: $\$30 \times 1.10 = \$33$.
Key warning: do NOT add $-25\%$ and $+10\%$ together. Multiply sequentially. The answer is C · $33.00.
04
Geometry · Right Triangle
★★★
A right triangle has legs of length 6 and 8. What is the area of the triangle?
📐 Solution
For a right triangle, the two legs are the base and height.
$A = \dfrac{1}{2} \times 6 \times 8 = \dfrac{1}{2} \times 48 = \mathbf{24}$
Note: the hypotenuse = $\sqrt{6^2+8^2} = 10$ (a 6-8-10 triple, scaled from 3-4-5), but you do not need it for area. The answer is C · 24.
$A = \dfrac{1}{2} \times 6 \times 8 = \dfrac{1}{2} \times 48 = \mathbf{24}$
Note: the hypotenuse = $\sqrt{6^2+8^2} = 10$ (a 6-8-10 triple, scaled from 3-4-5), but you do not need it for area. The answer is C · 24.
05
Probability · Complement
★★★
A bag contains 3 red, 4 blue, and 5 green marbles. If one marble is chosen at random, what is the probability that it is not red?
📐 Solution
Total marbles = $3 + 4 + 5 = 12$. Not-red = $4 + 5 = 9$.
$P(\text{not red}) = \dfrac{9}{12} = \dfrac{3}{4}$
Or use complement: $P(\text{not red}) = 1 - P(\text{red}) = 1 - \dfrac{3}{12} = 1 - \dfrac{1}{4} = \dfrac{3}{4}$. The answer is D · 3/4.
$P(\text{not red}) = \dfrac{9}{12} = \dfrac{3}{4}$
Or use complement: $P(\text{not red}) = 1 - P(\text{red}) = 1 - \dfrac{3}{12} = 1 - \dfrac{1}{4} = \dfrac{3}{4}$. The answer is D · 3/4.
06
Number Theory · Divisors
★★★
How many positive divisors does 72 have?
📐 Solution
$72 = 2^3 \times 3^2$
Number of divisors $= (3+1)(2+1) = 4 \times 3 = \mathbf{12}$
List: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. The answer is C · 12.
Number of divisors $= (3+1)(2+1) = 4 \times 3 = \mathbf{12}$
List: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. The answer is C · 12.
07
Algebra · Rate · Distance
★★★
Two trains are 280 miles apart and travel toward each other. Train A travels at 60 mph and Train B at 80 mph. How many hours will it take before they meet?
📐 Solution
When two objects move toward each other, combine their speeds.
Combined speed $= 60 + 80 = 140$ mph.
$t = \dfrac{280}{140} = \mathbf{2}$ hours. The answer is B · 2 hours.
Combined speed $= 60 + 80 = 140$ mph.
$t = \dfrac{280}{140} = \mathbf{2}$ hours. The answer is B · 2 hours.
08
Statistics · Mean & Median
★★★
A student scored 72, 85, 90, 68, and 95 on five tests. What is the mean score?
📐 Solution
Sum $= 72 + 85 + 90 + 68 + 95 = 410$.
Mean $= \dfrac{410}{5} = \mathbf{82}$.
Note: the median (middle value when sorted: 68, 72, 85, 90, 95) is 85, not 82. Don't confuse mean and median! The answer is C · 82.
Mean $= \dfrac{410}{5} = \mathbf{82}$.
Note: the median (middle value when sorted: 68, 72, 85, 90, 95) is 85, not 82. Don't confuse mean and median! The answer is C · 82.
09
Counting · Combinations
★★★
In how many ways can 2 students be chosen from a group of 6 to represent their class?
📐 Solution
Order does not matter (combination, not permutation).
$C(6,2) = \dfrac{6!}{2!\,4!} = \dfrac{6 \times 5}{2 \times 1} = \dfrac{30}{2} = \mathbf{15}$. The answer is B · 15.
$C(6,2) = \dfrac{6!}{2!\,4!} = \dfrac{6 \times 5}{2 \times 1} = \dfrac{30}{2} = \mathbf{15}$. The answer is B · 15.
10
Geometry · Circle
★★★
A circle has a radius of 7. What is its area? (Leave answer in terms of $\pi$.)
📐 Solution
$A = \pi r^2 = \pi (7)^2 = \mathbf{49\pi}$
Common mistake: using $2r = 14$ instead of $r^2 = 49$. The circumference is $2\pi r = 14\pi$, not the area. The answer is D · $49\pi$.
Common mistake: using $2r = 14$ instead of $r^2 = 49$. The circumference is $2\pi r = 14\pi$, not the area. The answer is D · $49\pi$.
11
Algebra · Arithmetic Sequence
★★★
The sequence 3, 7, 11, 15, … follows a pattern. What is the 20th term?
📐 Solution
$a_1 = 3$, common difference $d = 4$.
$a_{20} = 3 + (20 - 1) \times 4 = 3 + 76 = \mathbf{79}$
Watch out: $(20-1)$, not $(20)$. Many students use $n$ instead of $(n-1)$. The answer is C · 79.
$a_{20} = 3 + (20 - 1) \times 4 = 3 + 76 = \mathbf{79}$
Watch out: $(20-1)$, not $(20)$. Many students use $n$ instead of $(n-1)$. The answer is C · 79.
12
Ratio · Proportions
★★★
The ratio of boys to girls in a class is 3:5. If there are 24 boys, how many girls are there?
📐 Solution
Ratio boys:girls $= 3:5$. Since boys $= 24 = 3 \times 8$, the multiplier is $k = 8$.
Girls $= 5 \times 8 = \mathbf{40}$. The answer is C · 40.
Girls $= 5 \times 8 = \mathbf{40}$. The answer is C · 40.
13
Geometry · Triangle Angles
★★★
The three angles of a triangle are in the ratio $1:2:3$. What is the measure of the largest angle?
📐 Solution
Let angles be $x$, $2x$, $3x$. Sum = $180°$:
$x + 2x + 3x = 6x = 180° \Rightarrow x = 30°$
Largest angle $= 3x = 3 \times 30° = \mathbf{90°}$. The answer is C · 90°.
$x + 2x + 3x = 6x = 180° \Rightarrow x = 30°$
Largest angle $= 3x = 3 \times 30° = \mathbf{90°}$. The answer is C · 90°.
14
Percents · Percent Change
★★★
A town's population grew from 500 to 650. What is the percent increase?
📐 Solution
$\% \text{ increase} = \dfrac{650 - 500}{500} \times 100 = \dfrac{150}{500} \times 100 = \mathbf{30\%}$
Always divide by the original value, not the new one. The answer is C · 30%.
Always divide by the original value, not the new one. The answer is C · 30%.
15
Probability · Two Dice
★★★
Two fair six-sided dice are rolled. What is the probability that the sum of the two dice equals 7?
📐 Solution
Total outcomes $= 6 \times 6 = 36$.
Pairs summing to 7: $(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$ → 6 pairs.
$P = \dfrac{6}{36} = \dfrac{1}{6}$. Note: sum 7 is the most common sum on two dice. The answer is C · 1/6.
Pairs summing to 7: $(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$ → 6 pairs.
$P = \dfrac{6}{36} = \dfrac{1}{6}$. Note: sum 7 is the most common sum on two dice. The answer is C · 1/6.
16
Geometry · Rectangle
★★★
The perimeter of a rectangle is 48. The length is 3 times the width. What is the area of the rectangle?
📐 Solution
Let width $= w$, then length $= 3w$.
$P = 2(w + 3w) = 2(4w) = 8w = 48 \Rightarrow w = 6$, $l = 18$.
Area $= 6 \times 18 = \mathbf{108}$. The answer is C · 108.
$P = 2(w + 3w) = 2(4w) = 8w = 48 \Rightarrow w = 6$, $l = 18$.
Area $= 6 \times 18 = \mathbf{108}$. The answer is C · 108.
17
Fractions · Addition
★★★
What is the value of $\dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{6}$?
📐 Solution
LCD = 6: $\dfrac{3}{6} + \dfrac{2}{6} + \dfrac{1}{6} = \dfrac{3+2+1}{6} = \dfrac{6}{6} = \mathbf{1}$. The answer is C · 1.
18
Number Theory · GCD
★★★
What is the greatest common divisor (GCD) of 84 and 120?
📐 Solution
$84 = 2^2 \times 3 \times 7$ and $120 = 2^3 \times 3 \times 5$.
GCD = product of common primes at lowest power: $2^2 \times 3 = 4 \times 3 = \mathbf{12}$. The answer is C · 12.
GCD = product of common primes at lowest power: $2^2 \times 3 = 4 \times 3 = \mathbf{12}$. The answer is C · 12.
19
Algebra · Work Rate
★★★
A pipe can fill a pool in 4 hours, but a drain empties it in 12 hours. If both the pipe and drain are open at the same time, how many hours will it take to fill the empty pool?
📐 Solution
Fill rate $= \dfrac{1}{4}$ pool/hr. Drain rate $= \dfrac{1}{12}$ pool/hr.
Net rate $= \dfrac{1}{4} - \dfrac{1}{12} = \dfrac{3}{12} - \dfrac{1}{12} = \dfrac{2}{12} = \dfrac{1}{6}$ pool/hr.
Time $= \dfrac{1}{1/6} = \mathbf{6}$ hours. The answer is D · 6 hours.
Net rate $= \dfrac{1}{4} - \dfrac{1}{12} = \dfrac{3}{12} - \dfrac{1}{12} = \dfrac{2}{12} = \dfrac{1}{6}$ pool/hr.
Time $= \dfrac{1}{1/6} = \mathbf{6}$ hours. The answer is D · 6 hours.
20
Geometry · Similar Triangles
★★★
A triangle has sides of length 3, 4, and 5. A similar triangle has its shortest side equal to 9. What is the perimeter of the larger triangle?
📐 Solution
Scale factor $= \dfrac{9}{3} = 3$.
Original perimeter $= 3 + 4 + 5 = 12$.
Larger perimeter $= 12 \times 3 = \mathbf{36}$. The answer is D · 36.
Original perimeter $= 3 + 4 + 5 = 12$.
Larger perimeter $= 12 \times 3 = \mathbf{36}$. The answer is D · 36.
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