The Fundamental Theorem of Arithmetic & Factorization
Every integer greater than 1 can be represented uniquely as a product of prime numbers (disregarding their order). This cornerstone of number theory, proven by Euclid, implies that prime numbers serve as the fundamental "atoms" of all integers.
Number Classifications by Divisor Sum
| Classification | Definition / Proper Divisors Sum (s(n)) | Classic Example |
|---|---|---|
| Prime Number | Only two distinct positive divisors: 1 and itself | 2, 3, 5, 7, 11, 13, 17, 997 |
| Perfect Number | Sum of proper divisors equals the number: s(n) = n | 6 (1+2+3=6), 28 (1+2+4+7+14=28), 496 |
| Deficient Number | Sum of proper divisors is strictly less than the number: s(n) < n | 8 (1+2+4=7 < 8), 9, 10, all primes |
| Abundant Number | Sum of proper divisors exceeds the number: s(n) > n | 12 (1+2+3+4+6=16 > 12), 18, 20, 360 |
Frequently Asked Questions (FAQ)
Why is the number 1 neither prime nor composite?
By mathematical definition, a prime number must have exactly two distinct positive divisors (1 and itself). Since 1 has only one divisor, classifying it as prime would violate the uniqueness of prime factorization in the Fundamental Theorem of Arithmetic.
How is prime factorization used in modern technology?
Prime factorization is the computational backbone of modern public-key cryptography (such as RSA encryption). Multiplying two massive 1000-digit prime numbers together takes milliseconds, but factoring the resulting 2000-digit product back into its primes would take supercomputers thousands of years.