Divisibility & Primes
Divisibility rules, prime numbers, GCD, LCM, and the fundamental theorem of arithmetic
Key Concepts
- Every integer > 1 has a unique prime factorisation
- GCD and LCM are computed from the prime factorisation
- The Euclidean algorithm finds the GCD quickly
- The number of divisors comes from the exponents in the factorisation
Important Formulae
| GCD–LCM relation | gcd(a,b) × lcm(a,b) = a × b |
| Number of divisors | n = p₁^a · p₂^b ⇒ d(n) = (a+1)(b+1)… |
| Sum of divisors | σ(n) = Π (pᵢ^(aᵢ+1) − 1)/(pᵢ − 1) |
Quick Tips
- To test if n is prime, check divisibility only up to √n.
- Factor first — most divisibility problems collapse once n is factorised.
Sample Practice Questions
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Sum of all divisors of 12 =
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Answer: 28
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How many zeros at the end of 100! (100 factorial)?
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Answer: 24
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The number 2024 is divisible by:
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Answer: 8
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Which of these is a perfect number?
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Answer: 6
Practice Questions
Practise PRMO questions on Divisibility & Primes. Answers are revealed after each question.
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What are the key concepts in Divisibility & Primes for PRMO?
Every integer > 1 has a unique prime factorisation; GCD and LCM are computed from the prime factorisation; The Euclidean algorithm finds the GCD quickly; The number of divisors comes from the exponents in the factorisation.
Which formulae are important for Divisibility & Primes in PRMO?
GCD–LCM relation: gcd(a,b) × lcm(a,b) = a × b; Number of divisors: n = p₁^a · p₂^b ⇒ d(n) = (a+1)(b+1)…; Sum of divisors: σ(n) = Π (pᵢ^(aᵢ+1) − 1)/(pᵢ − 1).
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