In This Lesson Prime Numbers Fundamental Theorem of Arithmetic GCD & LCM 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 The Sieve of Eratosthenes Distribution of Primes 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 Prime Numbers A prime p > 1 has exactly two divisors: 1 and itself. The first primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …
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2 is the only even prime — every even number > 2 is divisible by 2. To test if n is prime, check divisibility by primes up to √n (why? if n = a·b, one factor must be ≤ √n).
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Fundamental Theorem of Arithmetic
Every integer n > 1 has a unique prime factorization:
n = p₁^a₁ · p₂^a₂ · … · pₖ^aₖ
Example: 360 = 2³ · 3² · 5 360 ÷ 2 = 180 → 180 ÷ 2 = 90 → 90 ÷ 2 = 45 → 45 ÷ 3 = 15 → 15 ÷ 3 = 5 → 5 ÷ 5 = 1
Divisors of 360: (3+1)(2+1)(1+1) = 24 divisors
GCD & LCM
GCD(a, b) = product of common primes with min exponents
LCM(a, b) = product of all primes with max exponents
a · b = GCD(a, b) · LCM(a, b)
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The Sieve of Eratosthenes To find all primes ≤ n: start with 2, mark all multiples of 2, next unmarked (3), mark all multiples of 3, continue to √n. The remaining unmarked numbers are prime. Complexity: O(n log log n).
Distribution of Primes The Prime Number Theorem : π(n) ≈ n/ln(n), where π(n) counts primes ≤ n. This connects primes to logarithmic functions and limits . There are infinitely many primes (Euclid's proof by contradiction is one of the most elegant in mathematics).
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