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What Is Prime Factorization?
Prime factorization expresses a number as a product of prime numbers. For example, 360 = 2³ × 3² × 5. Every integer greater than 1 has a unique prime factorization — this is the Fundamental Theorem of Arithmetic. Primes are the "building blocks" of all natural numbers.
How to Find Prime Factors
Start dividing by the smallest prime (2). When 2 no longer divides evenly, move to 3, then 5, 7, 11, and so on. Continue until the quotient is 1. For 360: 360÷2=180, 180÷2=90, 90÷2=45, 45÷3=15, 15÷3=5, 5÷5=1. So 360 = 2³ × 3² × 5.
Factor Trees
A factor tree is a visual method for finding prime factors. Start with the number and split it into any two factors. Continue splitting until all leaves are prime. Different trees (starting splits) always produce the same prime factors. For example, 360 → 36 × 10 → (6×6) × (2×5) → (2×3)(2×3)(2×5).
Applications of Prime Factorization
Prime factorization is used to find GCF and LCM, simplify fractions, determine the number of factors, and solve divisibility problems. In cryptography (RSA encryption), the difficulty of factoring large numbers into primes secures online communications and banking.
Frequently Asked Questions
No. By mathematical convention, 1 is neither prime nor composite. A prime number must have exactly two distinct factors (1 and itself). The number 1 has only one factor.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47... Note that 2 is the only even prime number. All other even numbers are divisible by 2.
For GCF, take the lowest power of each shared prime. For LCM, take the highest power of every prime. Example: 12 = 2²×3, 18 = 2×3². GCF = 2¹×3¹ = 6, LCM = 2²×3² = 36.
Finding prime factors of small-to-medium numbers is straightforward. For very large numbers (hundreds of digits), it becomes computationally difficult — this difficulty is the basis of RSA cryptographic security.