Understanding Prime Factorization and Divisors
Every whole number greater than 1 can be categorized as either a **prime number** or a **composite number**. A prime number has exactly two positive divisors: 1 and itself. A composite number, by contrast, has additional factors. The **prime factorization** is the unique combination of prime numbers that, when multiplied together, produce the original integer.
Base-Exponent Representation
Instead of listing repeating prime factors individually (e.g. 2 × 2 × 2 × 3), we write the factorization in standard mathematical form using exponents (e.g. 23 × 3). This compact notation shows each prime base raised to the power of its multiplicity.
Generating Divisors from Prime Factors
Knowing the prime factorization simplifies finding **all divisors** of a number. If an integer is factorized as n = p1a • p2b, any divisor of n will be formed by combining the prime bases raised to any power from 0 up to their respective exponents.
The **total count** of these divisors is determined by the formula: