Guides
You can count how many factors a number has without listing a single one: write its prime factorization, add 1 to each exponent, and multiply. This guide shows the formula, why it works, and what the count tells you.
Check any answer in the Factors Calculator.
The divisor-count formula
If n = p₁^a × p₂^b × p₃^c …, then the number of factors is (a+1)(b+1)(c+1)…. Start from the prime factorization.
Why it works
Each factor is built by choosing, for every prime, an exponent from 0 up to its maximum. A prime with exponent a offers a+1 choices, and the choices multiply.
Worked examples
36 = 2² × 3² → (2+1)(2+1) = 9 factors. 100 = 2² × 5² → 9 factors. 48 = 2⁴ × 3 → (4+1)(1+1) = 10 factors.
Perfect squares and primes
A perfect square always has an odd factor count (every exponent is even, so each +1 is odd). A prime has exactly two factors. A number with exactly three factors must be the square of a prime.
Quick checks
If the count comes out even, the number is not a perfect square. Use the formula to predict the count, then confirm with the Factors Calculator.
Key takeaways
- Factor count = product of (each exponent + 1).
- Perfect squares have an odd factor count.
- Primes have exactly two factors.
- Three factors means the square of a prime.
Factors Calculator
List every divisor and see the factor count, sum and pairs for any number.
Open the Factors CalculatorFrequently asked questions
How do you count factors without listing them?
Write the prime factorization, add 1 to each exponent, then multiply those results.
Why do perfect squares have an odd number of factors?
Their prime exponents are all even, so each (exponent + 1) is odd, and the product of odd numbers is odd.
What kind of number has exactly three factors?
The square of a prime, such as 49 = 7² (factors 1, 7, 49).
The LCM Calculator Team
Math educators and engineers building free, accurate calculators with step-by-step solutions, visual diagrams and AI insights.