Divisor Count Calculator

Divisor Count Calculator — Count Divisors Using Prime Factorization

Count the total number of divisors of any positive integer with step-by-step explanation. Learn the divisor function formula, understand how prime factorization determines divisor count, and explore the properties of the tau function. For related tools, see our factors of a number calculator.

Table of Contents

1. 1What Is the Divisor Count?
2. 2The Divisor Count Formula
3. 3How to Calculate Divisor Count
4. 4Step-by-Step Examples
5. 5More Divisor Count Examples
6. 6Divisor Count of Perfect Squares
7. 7The Divisor Function d(n)
8. 8Highly Composite Numbers
9. 9Common Mistakes to Avoid
10. 10Real-World Applications
11. 11Practice Problems with Solutions
12. 12Tips for Mastering Divisor Count
13. 13Final Thoughts
14. 14FAQ: Common Questions

Count the Divisors

Enter a positive integer to find how many divisors it has. Click Calculate for step-by-step working.

Enter a Number

Divisor count will appear here.

Note: The divisor function d(n) counts positive divisors. Using prime factorization n = p₁ᵃ¹ × p₂ᵃ² × ..., the formula is: d(n) = (a₁ + 1)(a₂ + 1)...(aₖ + 1).

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Divisor Count – Complete Explanation

The divisor count (also called the divisor function d(n) or τ(n)) tells you how many positive integers divide evenly into a given number. The formula is elegant: multiply one more than each exponent in the prime factorization.

Divisor Count Formula

If n = p₁ᵃ¹ × p₂ᵃ² × ... × pₖᵃᵏ

d(n) = (a₁ + 1)(a₂ + 1)...(aₖ + 1)

Example: 36 = 2² × 3² → d(36) = (2+1)(2+1) = 3 × 3 = 9.

1. What Is the Divisor Count?

The divisor count answers: "How many different numbers divide evenly into n?"

Key facts:

• d(1) = 1.
• d(p) = 2 for primes.
• d(n) is odd ⇔ n is a perfect square.
• d(n) is multiplicative for coprime factors.

2. The Divisor Count Formula Explained

For 36 = 2² × 3²:

Any divisor has form 2ˣ × 3ʸ where x ∈ {0,1,2}, y ∈ {0,1,2}.

3 choices × 3 choices = 9 divisors.

3. How to Calculate the Divisor Count

1. Prime factorize n.
2. Add 1 to each exponent.
3. Multiply the results.

Why add 1? For pᵉ, you can include p⁰, p¹, ..., pᵉ — that's (e+1) choices.

4. Step-by-Step Examples

Example 1: d(36)

36 = 2² × 3²
d(36) = (2+1)(2+1) = 3 × 3 = 9
Divisors: 1, 2, 3, 4, 6, 9, 12, 18, 36 ✓

Example 2: d(48)

48 = 2⁴ × 3¹
d(48) = (4+1)(1+1) = 5 × 2 = 10
Divisors: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 ✓

5. More Divisor Count Examples

d(12) = 6 — divisors: 1,2,3,4,6,12

d(100) = 9 — perfect square, odd count

d(17) = 2 — prime

d(1) = 1 — special case

6. Divisor Count of Perfect Squares

Perfect squares always have an odd divisor count:

36 = 2²×3² → d(36) = 9 (odd) ✓

100 = 2²×5² → d(100) = 9 (odd) ✓

144 = 2⁴×3² → d(144) = 15 (odd) ✓

7. The Divisor Function d(n)

d(n) = count of divisors (also τ(n))

σ(n) = sum of divisors

σₖ(n) = sum of k-th powers of divisors

8. Highly Composite Numbers

Numbers with more divisors than any smaller number:

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ...

9. Common Mistakes to Avoid

• Forgetting to add 1 to exponents.
• Adding instead of multiplying.
• Confusing d(n) with σ(n).
• Thinking primes have 1 divisor.
• Forgetting d(1) = 1.

10. Real-World Applications

• Arranging objects in rectangular grids — d(n) ways.
• Divisibility tests — verifying all factors found.
• Math competitions — divisor count problems.
• Computer science — algorithms for factorization.

11. Practice Problems with Solutions

Problem 1: d(72) = ? → 72 = 2³×3² → 12

Problem 2: d(120) = ? → 120 = 2³×3×5 → 16

Problem 3: Why does 64 have fewer divisors than 60?

Solution: d(64)=7, d(60)=12 — distinct primes create more divisors.

Problem 4: Find a number with exactly 6 divisors. → 12

12. Tips for Mastering Divisor Count

• Memorize: d(n) = Π(eᵢ + 1).
• Primes have 2 divisors.
• Perfect squares have odd d(n).
• More distinct primes → more divisors.

13. Final Thoughts

The divisor count is one of the most elegant functions in number theory. Use this calculator to verify your work, but practice applying the formula until it becomes automatic.

14. Frequently Asked Questions

The divisor count function d(n) (also called τ(n)) counts how many positive integers divide evenly into n. For example, d(12) = 6 because the divisors of 12 are 1, 2, 3, 4, 6, and 12.

If n = p₁ᵃ¹ × p₂ᵃ² × ..., then d(n) = (a₁+1)(a₂+1).... Add 1 to each exponent in the prime factorization and multiply. For 36 = 2² × 3², d(36) = (2+1)(2+1) = 9.

In a perfect square, all exponents are even, so each (exponent + 1) is odd. The product of odd numbers is odd. The square root pairs with itself as a divisor, counting once instead of twice — like 6 × 6 = 36, where 6 is one divisor, not two.

A prime number p has exactly 2 divisors: 1 and p. This is the definition of a prime — a number with exactly two positive divisors. So d(p) = 2 for all primes.

d(n) counts the number of divisors. σ(n) sums them. For 12: d(12) = 6 (six divisors), but σ(12) = 1+2+3+4+6+12 = 28. They are different functions answering different questions.

Highly composite numbers have more divisors than any smaller positive integer. The first few are 1, 2, 4, 6, 12, 24, 36, 48, 60. They're designed to maximize divisibility relative to their size.

d(1) = 1. The number 1 has exactly one divisor: itself. This is a special case. The prime factorization of 1 is empty, and the empty product gives 1.

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Related Calculators

Explore these other math tools that complement divisor count:

• 🔹 Factors of a Number Calculator — Find all factors and factor pairs.
• 🔹 Prime Factorization Calculator — Break numbers into prime factors.
• 🔹 Sum of Divisors Calculator — Calculate σ(n).
• 🔹 Perfect Number Checker — Check if a number is perfect.
• 🔹 Prime Number Checker — Check if a number is prime.
• 🔹 GCD Calculator — Find the greatest common divisor.

Conclusion

The divisor count d(n) is one of the most elegant functions in number theory. Use this calculator to verify your work, but practice applying the formula until it becomes automatic.

Readers who wish to continue practicing and reinforce what they've learned can explore the companion learning resource for guided exercises.