Find Factor Calculator

Find Factors of a Number

Enter a positive integer below to find all its factors, prime factors, and more using our Factor Calculator.

Table of factor pairs for the entered number.

Factor 1	Factor 2

What is a Factor Calculator?

A Factor Calculator is a tool used to find all the numbers (factors or divisors) that divide a given integer exactly, without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. Our Factor Calculator not only lists all factors but also provides the number of factors, the sum of factors, and the prime factors of the number.

This tool is useful for students learning about number theory, teachers preparing materials, and anyone needing to find the divisors of a number quickly. It helps in understanding the composition of numbers and concepts like prime factorization and divisibility.

Common misconceptions include thinking that factors are only prime numbers or that the number itself isn't a factor. Every positive integer has at least two factors: 1 and itself (except for 1, which has only one factor: 1). Our Factor Calculator clarifies these by listing all factors comprehensively.

Factor Finding Formula and Mathematical Explanation

To find the factors of a positive integer 'n', we look for all integers 'd' such that 'n / d' is also an integer. This means 'd' divides 'n' exactly.

The most straightforward method, used by our Factor Calculator, is to check every integer from 1 up to 'n'. If an integer 'i' divides 'n' with no remainder (n % i == 0), then 'i' is a factor.

A more efficient approach is to check integers from 1 up to the square root of 'n'. If 'i' divides 'n', then both 'i' and 'n/i' are factors. This is because if 'i' is a factor, and 'i > sqrt(n)', then 'n/i' must be less than 'sqrt(n)' and would have already been found (or 'i' is sqrt(n) itself).

For example, to find factors of 12:

1. Check 1: 12 / 1 = 12. Factors are 1 and 12.
2. Check 2: 12 / 2 = 6. Factors are 2 and 6.
3. Check 3: 12 / 3 = 4. Factors are 3 and 4.
4. Check 4: 12 / 4 = 3. We already found 3 and 4. Since 4 > sqrt(12) (approx 3.46), we can stop or realize we will find duplicates from here.

The factors are 1, 2, 3, 4, 6, 12. Our Factor Calculator automates this process.

The prime factorization of a number is expressing it as a product of its prime factors. For example, 12 = 2² * 3¹. The prime factors are 2 and 3.

Variables Involved:

Variable	Meaning	Unit	Typical Range
n	The integer whose factors are to be found	None (integer)	Positive integers (e.g., 1, 10, 1000)
d	A factor of n	None (integer)	1 to n

Practical Examples (Real-World Use Cases)

Example 1: Finding Factors of 36

Input Number: 36

Using the Factor Calculator:

• Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36
• Number of Factors: 9
• Sum of Factors: 91
• Prime Factors: 2, 3 (36 = 2² * 3²)
• Factor Pairs: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6)

This can be useful in mathematics for simplifying fractions or understanding the structure of 36.

Example 2: Finding Factors of 17

Input Number: 17

Using the Factor Calculator:

• Factors: 1, 17
• Number of Factors: 2
• Sum of Factors: 18
• Prime Factors: 17 (17 is a prime number)
• Factor Pairs: (1, 17)

Since 17 only has two factors (1 and itself), it is a prime number. Our Factor Calculator quickly identifies this.

How to Use This Factor Calculator

1. Enter the Number: Type the positive integer for which you want to find the factors into the input field labeled "Enter a positive integer".
2. Calculate: Click the "Calculate Factors" button or simply change the number (the calculator updates in real time if input is valid).
3. View Results: The calculator will display:
   • All factors of the number.
   • The total number of factors.
   • The sum of all factors.
   • The prime factors and their powers.
   • A table of factor pairs.
   • A chart of prime factors and their exponents (if applicable).
4. Reset: Click "Reset" to clear the input and results and start with the default value.
5. Copy: Click "Copy Results" to copy the main findings to your clipboard.

The results from the Factor Calculator help in understanding number properties and are foundational for topics like greatest common divisor (GCD) and least common multiple (LCM).

Key Factors That Affect Factor Calculator Results

The factors of a number are entirely determined by the number itself. Here are some properties of the input number that influence its factors:

1. Magnitude of the Number: Larger numbers generally have more factors, although not always. A large prime number will only have two factors.
2. Prime vs. Composite: Prime numbers have only two factors (1 and themselves). Composite numbers have more than two factors. Our Factor Calculator handles both.
3. Even vs. Odd: Even numbers always have 2 as a factor. Odd numbers do not.
4. Perfect Squares: Perfect squares (like 4, 9, 16, 25, 36) have an odd number of factors because one of their factor pairs consists of two identical numbers (e.g., 6×6=36).
5. Prime Factorization: The number and exponents of the prime factors determine the total number of factors. If a number n = p₁^a₁ * p₂^a₂ * … * p_k^a_k, the number of factors is (a₁+1)(a₂+1)…(a_k+1). The Factor Calculator shows this via the prime factors display and chart. Learn more about number theory basics.
6. Highly Composite Numbers: Some numbers have more factors than any smaller positive integer. These are called highly composite numbers (e.g., 1, 2, 4, 6, 12, 24, 36, 48, 60, 120).

Frequently Asked Questions (FAQ)

Q1: What are the factors of 1?

A1: The only factor of 1 is 1 itself.

Q2: What are the factors of 0?

A2: Technically, every non-zero integer is a factor of 0 because 0 divided by any non-zero integer is 0. However, our Factor Calculator is designed for positive integers.

Q3: How many factors does a prime number have?

A3: A prime number has exactly two factors: 1 and the number itself.

Q4: Can a number have an odd number of factors?

A4: Yes, perfect squares have an odd number of factors. For example, 9 has factors 1, 3, 9 (three factors).

Q5: Does the Factor Calculator find negative factors?

A5: No, this Factor Calculator focuses on positive factors of positive integers. If 'd' is a positive factor of 'n', then '-d' is also a factor, but typically only positive factors are listed.

Q6: What is the difference between factors and prime factors?

A6: Factors are any numbers that divide the given number exactly. Prime factors are the prime numbers that, when multiplied together, produce the original number. For 12, factors are 1, 2, 3, 4, 6, 12; prime factors are 2, 3 (since 12 = 2x2x3).

Q7: How is this Factor Calculator related to divisibility rules?

A7: Divisibility rules are shortcuts to check if a number is divisible by small integers (like 2, 3, 4, 5, 6, 9, 10). Finding factors involves checking for divisibility systematically.

Q8: Can I use this Factor Calculator for very large numbers?

A8: The calculator works best for reasonably sized integers. Very large numbers (e.g., with hundreds of digits) require specialized algorithms beyond this tool's scope due to computational limits in web browsers.

Related Tools and Internal Resources

• Prime Factorization Calculator: Find the prime factors of any number.
• Greatest Common Divisor (GCD) Calculator: Find the largest number that divides two integers.
• Least Common Multiple (LCM) Calculator: Find the smallest number that is a multiple of two integers.
• Divisibility Rules Guide: Learn quick rules to check for divisibility by common numbers.
• Number Theory Basics: An introduction to the properties of integers.
• Integer Properties Explorer: Discover more about different types of integers.