Number of factors = (a+1)(b+1)(c+1)…
How many factors does 360 have?
360 = 2³ · 3² · 5¹
→ (3+1)(2+1)(1+1) = 24 factors
Product of roots = c/a
x + y = 10, xy = 21. Find x² + y².
→ 10² − 2(21) = 100 − 42 = 58
Distance = √[(x₂−x₁)² + (y₂−y₁)²]
Right triangle with legs 5 and 12. Find the hypotenuse.
→ √(25+144) = √169 = 13
C(n,r) = n! / [r!(n−r)!]
P(B|A) = P(A∩B) / P(A)
Arrange 4 books chosen from 6 different books. How many ways?
→ P(6,4) = 6×5×4×3 = 360
Sₙ = n(a₁ + aₙ)/2
Sₙ = a₁(rⁿ − 1)/(r − 1)
Geometric: a₁ = 2, r = 3. Find S₄.
→ 2(3⁴−1)/(3−1) = 2(80)/2 = 80
Ready to compete?
Answer all 20 questions. The timer starts when you begin.
Correct answers earn a confetti moment. Wrong answers reveal the solution immediately.
What is the sum of all positive integers less than 20 that are divisible by neither 2 nor 3?
List all integers from 1 to 19 not divisible by 2 or 3:
Sum = 1+5+7+11+13+17+19 = 73
These are exactly the integers coprime to 6 in the range.
A store first marks up an item by 25%, then applies a 20% discount to the new price. What is the net percentage change from the original price?
Let original price = $100.
After 20% discount: 125 × 0.80 = $100
Net change = 100 − 100 = 0%
Key insight: 1.25 × 0.80 = 1.00. The two operations exactly cancel.
A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?
Apply the Pythagorean theorem:
c = √169 = 13
(5, 12, 13) is a fundamental Pythagorean triple. Memorize it!
How many 3-digit positive integers have digits that sum to exactly 5?
The hundreds digit h ≥ 1. Let h go from 1 to 5; tens digit t and units digit u = 5 − h − t ≥ 0.
h=2: t=0..3 → 4 numbers
h=3: t=0..2 → 3 numbers
h=4: t=0..1 → 2 numbers
h=5: t=0 → 1 number
Total = 5+4+3+2+1 = 15
In an arithmetic sequence, the first term is 3 and the common difference is 7. What is the 10th term?
a₁₀ = 3 + (10−1)(7)
= 3 + 9 × 7
= 3 + 63 = 66
If x + y = 10 and xy = 21, what is the value of x² + y²?
Use the key identity: x² + y² = (x+y)² − 2xy
= 100 − 42
= 58
What is the least common multiple (LCM) of 12 and 18?
18 = 2 × 3²
LCM = 2² × 3² = 4 × 9 = 36
Take the highest power of each prime factor.
A circle has a radius of 5. What is the area of the circle in terms of π?
A common error is using 2r or forgetting to square. Always: A = πr².
Events A and B satisfy P(A) = 0.4, P(B) = 0.5, and P(A ∩ B) = 0.2. What is P(A ∪ B)?
Inclusion-Exclusion Principle:
= 0.4 + 0.5 − 0.2
= 0.7
A train travels 240 miles in 4 hours. At the same speed, how many hours will it take to travel 360 miles?
Time = 360 ÷ 60 = 6 hours
Or use proportions: 240/4 = 360/t → t = 360×4/240 = 6.
In how many ways can 4 books be selected and arranged in order from a shelf of 6 different books?
Order matters, so use permutations:
= 6 × 5 × 4 × 3
= 360
A rectangle has length 8 and width 6. What is the length of its diagonal?
d = √100 = 10
(6, 8, 10) = 2 × (3, 4, 5). Recognize scaled Pythagorean triples!
What are the solutions to x² − 7x + 12 = 0?
Factor: find two numbers with product 12 and sum 7.
x = 3 or x = 4
Verify: 3+4=7 ✓, 3×4=12 ✓
How many positive divisors does 360 have?
Number of divisors = (3+1)(2+1)(1+1)
= 4 × 3 × 2 = 24
A triangle with sides 3, 4, and 5 is similar to a larger triangle whose shortest side has length 9. What is the perimeter of the larger triangle?
Larger sides = 9, 12, 15
Perimeter = 9 + 12 + 15 = 36
Note: 9, 12, 15 is 3 × (3,4,5) — still a right triangle.
What is the solution set of the inequality |2x − 3| ≤ 7?
⟹ −7 ≤ 2x − 3 ≤ 7
⟹ −4 ≤ 2x ≤ 10
⟹ −2 ≤ x ≤ 5
In a geometric sequence with first term 2 and common ratio 3, what is the sum of the first 4 terms?
Sₙ = a₁(rⁿ−1)/(r−1) = 2(3⁴−1)/(3−1)
= 2(81−1)/2 = 80
Check: 2+6+18+54 = 80 ✓
If P(A ∩ B) = 0.3 and P(A) = 0.5, what is P(B | A)?
Conditional probability formula:
= 0.3 / 0.5
= 0.6
What is the remainder when 2¹⁰ is divided by 7?
Find the repeating cycle of powers of 2 mod 7:
2² mod 7 = 4
2³ mod 7 = 8 mod 7 = 1 ← cycle length 3
10 = 3×3 + 1, so 2¹⁰ ≡ 2¹ = 2 (mod 7)
What is the midpoint of the segment connecting the points (2, 4) and (8, 10)?
= ((2+8)/2 , (4+10)/2)
= (10/2 , 14/2)
= (5, 7)