Greatest Prime Factor — Definition, Formula & Examples

The greatest prime factor of a whole number is the largest prime number that divides it evenly. For example, the prime factors of 30 are 2, 3, and 5, so its greatest prime factor is 5.

For any integer $n > 1$ with prime factorization $n = p_1^{a_1} \cdot p_2^{a_2} \cdots p_k^{a_k}$ where $p_1 < p_2 < \cdots < p_k$ , the greatest prime factor of $n$ is $p_k$ .

How It Works

To find the greatest prime factor, first break the number down into its prime factorization using a factor tree or repeated division. Then identify the largest prime in that factorization. The exponent on that prime does not matter — you only need to find which prime is biggest.

Worked Example

Problem: Find the greatest prime factor of 84.

Step 1: Divide by the smallest prime, 2.

84 \div 2 = 42

Step 2: Continue dividing by primes.

42 \div 2 = 21, \quad 21 \div 3 = 7

Step 3: Write the full prime factorization and identify the largest prime.

84 = 2^2 \times 3 \times 7

Answer: The greatest prime factor of 84 is 7.

Why It Matters

Identifying the greatest prime factor strengthens your ability to work with prime factorization, a skill used throughout algebra and number theory. Standardized math competitions and contests frequently ask students to find greatest prime factors quickly.

Common Mistakes

Mistake: Choosing the largest factor instead of the largest prime factor. For 84, a student might say 42 is the greatest prime factor.

Correction: 42 is a factor of 84 but is not prime (42 = 2 × 3 × 7). Only prime numbers qualify, so the answer is 7.