Distinct Prime Factors — Definition, Formula & Examples

Distinct prime factors are the different prime numbers that divide evenly into a given integer. For example, 12 has the prime factorization

2^2 \times 3

, so its distinct prime factors are 2 and 3.

For a positive integer $n > 1$ , the distinct prime factors of $n$ are the elements of the set $\{p : p \text{ is prime and } p \mid n\}$ . The number of distinct prime factors is denoted $\omega(n)$ in number theory.

Key Formula

\omega(n) = \text{number of distinct primes } p \text{ such that } p \mid n

Where:

$\omega(n)$ = Count of distinct prime factors of n
$n$ = A positive integer greater than 1
$p$ = A prime number that divides n

How It Works

Start by finding the prime factorization of the number. Then list each prime that appears, ignoring how many times it repeats. The count of those primes gives you the number of distinct prime factors. For instance,

72 = 2^3 \times 3^2

uses the prime 2 three times and the prime 3 twice, but it has only 2 distinct prime factors: 2 and 3.

Worked Example

Problem: Find all the distinct prime factors of 180.

Step 1: Find the prime factorization of 180.

180 = 2^2 \times 3^2 \times 5

Step 2: List each prime that appears, without repeats.

\{2,\, 3,\, 5\}

Step 3: Count them to find the number of distinct prime factors.

\omega(180) = 3

Answer: The distinct prime factors of 180 are 2, 3, and 5. There are 3 of them.

Why It Matters

Knowing the distinct prime factors of two numbers tells you quickly whether they are relatively prime (they share no common prime factors). This concept also appears when simplifying fractions and computing the GCF or LCM of large numbers.

Common Mistakes

Mistake: Counting repeated prime factors as separate distinct primes. For example, saying 8 has three prime factors because

8 = 2 \times 2 \times 2

Correction: Distinct means unique. Since 2 is the only prime involved,

\omega(8) = 1

. The exponent tells you multiplicity, not distinctness.