Prime factorization is the process of breaking down a composite number into a product of its prime factors. A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself. For example, the prime factorization of 72 is 2³ × 3², meaning 72 = 2 × 2 × 2 × 3 × 3. Every integer greater than 1 has a unique prime factorization — this is known as the Fundamental Theorem of Arithmetic.

A factor tree is a visual diagram that shows the step-by-step breakdown of a number into its prime factors. Starting from the original number at the top (the "root"), you repeatedly split each composite number into two factors. Branches end when all remaining numbers are prime (the "leaves"). The factor tree makes it easy to see how a number is constructed from primes, and it's an excellent educational tool for students learning about factors and multiples.

No, 1 is not a prime number. By definition, a prime number must have exactly two distinct positive divisors: 1 and itself. The number 1 only has one divisor (itself), so it does not qualify. It is also not a composite number. In modern mathematics, 1 is classified as a unit — it's neither prime nor composite. This is important for the Fundamental Theorem of Arithmetic, which requires that prime factorizations be unique.

Prime factorization plays a crucial role in RSA encryption, one of the most widely used public-key cryptosystems. RSA relies on the fact that multiplying two large prime numbers is computationally easy, but factoring their product back into the original primes is extremely difficult when the primes are large enough (e.g., 2048-bit numbers). This one-way function property makes prime factorization the foundation of secure online communication, including HTTPS, digital signatures, and email encryption.

The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. This means there is only one set of prime factors for any given number. For example, 84 = 2² × 3 × 7 — no other combination of primes will produce 84. This theorem is foundational to number theory and underpins much of modern mathematics.

To check if a number n is prime, you can use trial division: test divisibility by all numbers from 2 up to √n. If none divide evenly, n is prime. For small numbers this is fast and practical. For very large numbers, more sophisticated primality tests like the Miller-Rabin test are used. Some quick tips: all primes greater than 2 are odd, all primes greater than 3 are of the form 6k±1, and any number ending in 0, 2, 4, 5, 6, or 8 (except 2 and 5) is not prime.

A prime number has exactly two distinct positive divisors (1 and itself). Examples: 2, 3, 5, 7, 11, 13. A composite number has more than two positive divisors. Examples: 4 (divisors: 1, 2, 4), 6 (1, 2, 3, 6), 72 (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72). The number 1 is special — it's neither prime nor composite. Every composite number can be expressed as a unique product of prime factors.