In This Lesson Prime Numbers Fundamental Theorem of Arithmetic GCD & LCM 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, …
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).
Primes are the foundation of modern cryptography and connect deeply to factoring in algebra.
Fundamental Theorem of Arithmetic 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
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
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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)
Euclidean Algorithm computes GCD efficiently: GCD(a, b) = GCD(b, a mod b). This is one of the oldest algorithms — and it's essential in modular arithmetic for finding modular inverses.
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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