Factor Calculator
Find all factors, factor pairs, prime factorization, and interactive factor trees of any positive or negative integer.
Factor Calculator
Please provide an integer to calculate its factors, factor pairs, prime factorization, and interactive factor tree.
What is a Factor in Mathematics?
In number theory, a factor (or divisor) of an integer n is an integer d that divides n completely without leaving a remainder.
For example, 12 can be factored as 1 × 12, 2 × 6, and 3 × 4. Therefore, the positive factors of 12 are 1, 2, 3, 4, 6, and 12.
Prime Numbers vs. Composite Numbers
Prime Numbers
A prime number is an integer greater than 1 that has exactly two distinct positive divisors: 1 and the number itself.
Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 97...
Composite Numbers
A composite number is an integer greater than 1 that has more than two positive divisors. It can be factored into smaller positive integers.
Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 120...
How to Build a Factor Tree
A factor tree is a visual branching diagram used to break down any composite integer into its terminal prime factors:
- Write down the starting composite number at the top.
- Find any two factors whose product equals the number (e.g. for 120, choose 2 and 60).
- Draw two branches down to these factors.
- If a factor is prime, circle it (it is a terminal leaf).
- If a factor is composite, continue branching it until only prime numbers remain at the bottom.
Divisor Formulas & Classification
Derived from the exponents of its prime factor decomposition p₁^(e₁) × p₂^(e₂)...
Computes the exact arithmetic sum of all positive integer divisors.
Frequently Asked Questions
Is 1 considered a prime number or composite number?
Neither. In modern mathematics, 1 is defined as a unit (the multiplicative identity). Prime numbers must have exactly two distinct positive divisors, whereas 1 only has one (itself).
Why do perfect squares have an odd number of factors?
Factors of non-square numbers always appear in distinct pairs (a × b = n with a ≠ b). For a perfect square, one of the pairs consists of the square root multiplied by itself (k × k = n), resulting in an odd total count of divisors.
What is a perfect number?
A perfect number is a positive integer equal to the sum of its proper divisors (its aliquot sum). For example, 6 has proper divisors 1, 2, and 3, and 1 + 2 + 3 = 6.
Understanding Factor Calculator & Formula
The Factor Calculator on CalculatorScale.com is engineered to deliver instant, mathematically verified calculations conforming strictly to standard institutional equations. Whether planning long-term investments, managing household budgets, tracking clinical fitness targets, analyzing academic genetics, calculating veterinary dosages, or converting dimensional units, every calculation updates dynamically in real time without lag.
Every formula used in this tool adheres to certified international standards, including World Health Organization biometric metrics, veterinary clinical guidelines, NIST physical conversion constants, and standard biological equations. Results are computed entirely client-side inside your browser with complete data privacy, ensuring your sensitive numbers and statistics never leave your device.
Our calculators are 100% free with no registration, subscriptions, or paywalls required. You can toggle between imperial and metric units, customize input parameters, export calculation summaries, or embed this calculator directly into your own website or blog.
How to Use the Factor Calculator
- 1 Enter any positive or negative whole integer into the input field (e.g. 120).
- 2 Click the green "Calculate ▶️" button.
- 3 View all positive divisors, factor count, and factor pairs.
- 4 Explore the canonical exponential prime factorization, visual hierarchical factor tree diagram, and sum of divisors.
Factor Calculator — Frequently Asked Questions
Q1 What is the difference between factors and prime factors? ▾
Factors are all positive integers that divide a number evenly without a remainder. Prime factors are the subset of those factors that are also prime numbers, which can be multiplied together to produce the original number.
Q2 Why does 0 have infinitely many factors? ▾
Because any non-zero integer multiplied by zero equals zero (k × 0 = 0). Therefore, every integer is a divisor of 0.
Q3 How do you find the total number of factors using prime factorization? ▾
Take the exponent of each prime factor in the canonical prime factorization, add 1 to each exponent, and multiply these values together: d(n) = (e₁ + 1)(e₂ + 1)...(eₖ + 1).
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