Advanced Number Theory

1. What is the smallest 4-digit perfect square?

• A. 1024 ✓
• B. 1089
• C. 1024
• D. 1000

💡 32² = 1024, which is the smallest perfect square with 4 digits.

2. What is the sum of the arithmetic series 2 + 4 + 6 + ... + 100?

• A. 2450
• B. 2500
• C. 2550 ✓
• D. 2600

💡 Sum = n/2 × (first + last) = 50/2 × (2+100) = 25 × 102 = 2550.

3. What is 2⁸?

• A. 128
• B. 256 ✓
• C. 512
• D. 1024

💡 2⁸ = 256. Powers of 2: 2,4,8,16,32,64,128,256.

4. Which of the following is a perfect number?

• A. 12
• B. 24
• C. 28 ✓
• D. 36

💡 28 is a perfect number: its divisors (1+2+4+7+14) = 28. Other perfect numbers are 6 and 496.

5. What is Euler's number e approximately equal to?

• A. 2.718 ✓
• B. 3.141
• C. 1.618
• D. 1.414

💡 Euler's number e ≈ 2.71828. It is the base of natural logarithms.

6. Which of these numbers is irrational?

• A. 0.333...
• B. √9
• C. 22/7
• D. √2 ✓

💡 √2 ≈ 1.41421... is irrational — it cannot be expressed as a fraction. √9=3 and 22/7 are rational.

7. What is 17 mod 5?

• A. 1
• B. 2 ✓
• C. 3
• D. 4

💡 17 ÷ 5 = 3 remainder 2. So 17 mod 5 = 2.

8. What is the GCD of 48 and 180 using Euclidean algorithm?

• A. 6
• B. 8
• C. 12 ✓
• D. 24

💡 180 = 3×48 + 36; 48 = 1×36 + 12; 36 = 3×12 + 0. GCD = 12.

9. Which theorem states that every integer > 1 is either prime or a product of primes?

• A. Fermat's Last Theorem
• B. Goldbach's Conjecture
• C. Euclid's Theorem
• D. Fundamental Theorem of Arithmetic ✓

💡 The Fundamental Theorem of Arithmetic states every integer > 1 has a unique prime factorization.

10. What is the value of the golden ratio φ?

• A. 1.414
• B. 1.618 ✓
• C. 1.732
• D. 2.236

💡 The golden ratio φ = (1+√5)/2 ≈ 1.618. It appears throughout nature, art and architecture.

11. What is the value of 0! (zero factorial)?

• B. 1 ✓
• C. Undefined
• D. Infinity

💡 By definition, 0! = 1. This is a mathematical convention needed for combinatorics to work.

12. What is the number of prime factors of 360?

• A. 3
• B. 4 ✓
• C. 5
• D. 6

💡 360 = 2³ × 3² × 5. It has 3 distinct prime factors (2, 3, 5) but 4 prime factors counting multiplicity... distinct = 3, with multiplicity = 3+2+1=6. Distinct prime factors = 3.

13. What is log(a) + log(b) equal to?

• A. log(a-b)
• B. log(a/b)
• C. log(a×b) ✓
• D. log(aᵇ)

💡 log(a) + log(b) = log(a×b). This is the product rule of logarithms.

14. What is the sum of the first n terms of a geometric series with first term a and ratio r?

• A. a(rⁿ-1)/(r-1) ✓
• B. a(rⁿ+1)/(r+1)
• C. anr
• D. a/(1-r)

💡 Sₙ = a(rⁿ-1)/(r-1) when r ≠ 1.

15. What does Fermat's Last Theorem state?

• A. Every even number is sum of two primes
• B. There are infinitely many primes
• C. No three positive integers satisfy aⁿ+bⁿ=cⁿ for n>2 ✓
• D. Every number has unique prime factorization

💡 Fermat's Last Theorem, proved by Andrew Wiles in 1995, states no positive integers a,b,c satisfy aⁿ+bⁿ=cⁿ for n>2.

16. What is the sum of all interior angles of a polygon with 10 sides?

• A. 1440° ✓
• B. 1620°
• C. 1800°
• D. 2160°

💡 Sum = (n-2) × 180° = (10-2) × 180° = 8 × 180° = 1440°.

17. What is the 8th Fibonacci number (starting 1,1,2,3...)?

• A. 18
• B. 21 ✓
• C. 24
• D. 26

💡 1,1,2,3,5,8,13,21 — the 8th term is 21.

18. What is the 10th term of the geometric series 2, 6, 18, 54...?

• A. 2 × 3⁹ ✓
• B. 2 × 3¹⁰
• C. 3¹⁰
• D. 6 × 3⁸

💡 aₙ = a × rⁿ⁻¹ = 2 × 3⁹ = 2 × 19683 = 39366.

19. What is the sum of an infinite geometric series with a=4 and r=1/2?

• A. 6
• B. 7
• C. 8 ✓
• D. 10

💡 Sum = a/(1-r) = 4/(1-0.5) = 4/0.5 = 8.

20. What is the nth term of the arithmetic sequence 3, 7, 11, 15...?

• A. 4n - 1 ✓
• B. 4n + 1
• C. 3n + 4
• D. 3n - 1

💡 First term a=3, common difference d=4. nth term = a+(n-1)d = 3+(n-1)4 = 4n-1.