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
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How many positive divisors does 36 have?
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Answer: 9
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What is the last digit of 7⁴?
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Answer: 1
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What is the GCD of 48 and 60?
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Answer: 12
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Is the integer 0 even or odd?
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Answer: Even
Practice Questions
Practise RMO questions on Number Theory. Answers are revealed after each question.
Start Practice →Frequently Asked Questions
What are the key concepts in Number Theory for RMO?
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.
Which formulae are important for Number Theory in RMO?
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.
How can I practise RMO Number Theory questions for free?
Practise free RMO Number Theory questions with instant answers and explanations on Safe Learning for Kids — no sign-up required.