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What Is Prime Factorization?

Prime factorization is the process of expressing a composite number as a product of its prime factors. A prime number is a natural number greater than 1 that has no divisors other than 1 and itself (2, 3, 5, 7, 11, 13, ...). A composite number is any integer greater than 1 that is not prime.

The Fundamental Theorem of Arithmetic guarantees that every integer greater than 1 has a unique prime factorization (up to the order of factors). This uniqueness is why prime factorization is such a powerful tool. For example, 360 = 2³ × 3² × 5, and this is the only way to write 360 as a product of primes.

Our algorithm works by trial division: starting from 2, we check if the number is divisible by each integer up to its square root. Each time a factor is found, it's extracted and the process continues with the quotient. This is efficient for numbers up to tens of billions.

Applications of Prime Factorization

Prime factorization has numerous practical applications. Finding the GCD (Greatest Common Divisor): to find GCD(24, 36), factorize both — 24 = 2³ × 3, 36 = 2² × 3² — then take the minimum exponent of each prime: GCD = 2² × 3 = 12. Finding the LCM (Least Common Multiple): take the maximum exponent of each prime: LCM(24, 36) = 2³ × 3² = 72.

In cryptography, the difficulty of factoring large numbers (products of two large primes) is the foundation of RSA encryption, which secures most internet communications. The best known algorithms for factoring large numbers (general number field sieve) still require supercomputer time for 2048-bit numbers.

Prime factorization also appears in simplifying fractions (divide numerator and denominator by GCD), understanding number patterns, and solving divisibility problems. It's a cornerstone of number theory with deep connections to the distribution of primes (described by the Riemann Hypothesis).

How to Find Prime Factors Manually

The factor tree method is the most common manual approach. Start with the number and repeatedly branch it into two factors until all branches end in primes. For 180: 180 = 4 × 45 = 2 × 2 × 9 × 5 = 2 × 2 × 3 × 3 × 5. Collect all leaf nodes: 2² × 3² × 5.

The division method is more systematic: start dividing by 2 repeatedly until the number is odd, then try 3, 5, 7, and so on. For 360: 360 ÷ 2 = 180 ÷ 2 = 90 ÷ 2 = 45 ÷ 3 = 15 ÷ 3 = 5. So 360 = 2³ × 3² × 5.

A useful shortcut: you only need to check prime divisors up to the square root of the number. If no prime up to √n divides n, then n itself is prime. This reduces the work significantly for large numbers. For n = 100, you only need to check 2, 3, 5, 7 (since √100 = 10).

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