What is prime factorization?+
Prime factorization expresses a positive integer as a product of prime numbers. Every integer > 1 has exactly one prime factorization (Fundamental Theorem of Arithmetic). Example: 360 = 2³ × 3² × 5. The primes 2, 3, and 5 are the only prime building blocks of 360, appearing 3, 2, and 1 times respectively.
How do you find prime factorization step by step?+
Use trial division: (1) Start dividing by 2 — if it divides, record it and continue with the quotient. (2) Move to 3, then 5, then 7, etc. Only test primes up to √n. (3) When the quotient becomes 1, stop. Example for 180: 180÷2=90, 90÷2=45, 45÷3=15, 15÷3=5, 5 is prime. Result: 180 = 2² × 3² × 5.
What is the prime factorization of 360?+
360 = 2³ × 3² × 5. Division steps: 360÷2=180, 180÷2=90, 90÷2=45, 45÷3=15, 15÷3=5, 5 is prime. The number 360 has (3+1)(2+1)(1+1) = 24 divisors, which is why it was chosen by ancient Babylonians for angle measurement — 360 degrees divides evenly into 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 parts.
How do you find GCF and LCM using prime factorization?+
GCF: factorize both numbers, take shared prime factors with the LOWER exponent. LCM: take all prime factors with the HIGHER exponent. Example: GCF(12,18): 12=2²×3, 18=2×3² → GCF = 2¹×3¹ = 6. LCM = 2²×3² = 36. Shortcut: LCM = a×b/GCF = 12×18/6 = 36.
How many divisors does a number have?+
If n = p₁^e₁ × p₂^e₂ × ... × pₖ^eₖ, then divisors count = (e₁+1)(e₂+1)...(eₖ+1). Example: 360 = 2³×3²×5¹ → (3+1)(2+1)(1+1) = 24. For a prime p: p = p¹ → (1+1) = 2 divisors (1 and p). For a prime squared p²: (2+1) = 3 divisors (1, p, p²).
Is 1 a prime number?+
No. 1 is neither prime nor composite. By definition, primes have exactly two distinct factors (1 and themselves), but 1 has only one factor. Excluding 1 from primes preserves the Fundamental Theorem of Arithmetic's uniqueness — if 1 were prime, then 6 could be written as 2×3 or 1×2×3 or 1×1×2×3, losing unique factorization.
What is the Fundamental Theorem of Arithmetic?+
Every integer greater than 1 is either prime or can be expressed as a unique product of primes (up to reordering). This theorem has two parts: existence (every integer has at least one prime factorization) and uniqueness (there is exactly one such factorization). It is the foundation of number theory and makes GCF, LCM, and divisor counting well-defined operations.
What is the largest prime factor of 100?+
100 = 2² × 5². The prime factors are 2 and 5. The largest prime factor of 100 is 5. For comparison: the largest prime factor of 99 (= 3² × 11) is 11, and 101 is itself prime. The largest prime factor grows slowly: for n = 1000, it is 5 (since 1000 = 2³×5³).
How is prime factorization used in cryptography?+
RSA encryption secures internet transactions using the fact that multiplying two large primes (e.g., 1024-bit primes) is fast, but factoring their product back into the original primes is computationally infeasible with current technology. Your bank uses prime factorization's hardness every time you make an online payment — the encryption key is a product of two secret primes.
How do I find GCF without prime factorization?+
Use the Euclidean algorithm: repeatedly divide the larger by the smaller, keeping only the remainder, until the remainder is 0. The last non-zero remainder is the GCF. Example: GCF(48, 18): 48 = 2×18+12 → 18 = 1×12+6 → 12 = 2×6+0. GCF = 6. This is faster than factorization for very large numbers.