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Number Theory

Divisibility, primes, congruences, Diophantine equations, and number-theoretic functions

Key Concepts
  • Divisibility, primes and the unique factorisation theorem
  • Congruences and complete/reduced residue systems
  • Fermat, Euler and the Chinese Remainder Theorem
  • Diophantine equations and bounding techniques
Important Formulae
Euler's theorem a^φ(n) ≡ 1 (mod n), gcd(a,n)=1
Euler's totient φ(n) = n·Π(1 − 1/pᵢ)
GCD–LCM relation gcd(a,b) × lcm(a,b) = a × b
Quick Tips
  • Use congruences to restrict the possible cases before solving.
  • Infinite descent and bounding arguments prove many 'no solution' results.
Sample Practice Questions

  1. What is the sum of the first five prime numbers (2 + 3 + 5 + 7 + 11)?

    • 26
    • 28
    • 30
    • 18
    Show answer

    Answer: 28

  2. How many positive divisors does 36 have?

    • 6
    • 8
    • 9
    • 12
    Show answer

    Answer: 9

  3. Is the integer 0 even or odd?

    • Even
    • Odd
    • Neither
    • Both
    Show answer

    Answer: Even

  4. What is the last digit of 7⁴?

    • 1
    • 3
    • 7
    • 9
    Show answer

    Answer: 1

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Practice Questions

Practise RMO questions on Number Theory. Answers are revealed after each question.

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Number Theory