How to Use This Factoring Calculator

To get started, simply type a whole number (positive or negative) into the input field.

This tool provides three distinct layers of analysis:

1. List of Factors: Every integer that divides evenly into your number.
2. Factor Pairs: The specific couples of numbers that multiply together to reach your input.
3. Prime Factorization: The number broken down into its “prime DNA” (e.g., 12 = 2 x 2 x 3).

What is a Factor?

In mathematics, a factor (or divisor) is a number that divides another number completely, leaving no remainder.

Think of factors as the building blocks of a number. If we take the number 12, we can divide it evenly by 1, 2, 3, 4, 6, and 12. Therefore, 1, 2, 3, 4, 6, and 12 are the factors of 12.

Factors vs. Multiples

These terms are often confused, but the difference is simple directionality:

• Factors go into the number (they are smaller or equal to it).
• Multiples come out of the number (they are larger or equal to it).

Example:

• Factors of 10: 1, 2, 5, 10.
• Multiples of 10: 10, 20, 30, 40…

Factor Pairs: The “Rainbow” Method

Factors don’t exist in isolation; they essentially work in teams. A factor pair is a set of two integers that, when multiplied, result in the original number.

For the number 24, the pairs are:

• 1 x 24
• 2 x 12
• 3 x 8
• 4 x 6

When you use the calculator above, the “Factor Pairs” section organizes these into a clear multiplication table. This is particularly useful for students learning the “Rainbow Method,” where you connect the smallest factor to the largest, the second smallest to the second largest, and so on, creating a rainbow shape.

Prime Factorization: The DNA of Numbers

While a standard list of factors tells you what numbers fit inside, Prime Factorization tells you how the number is built.

According to the Fundamental Theorem of Arithmetic, every integer greater than 1 is either a prime number itself or can be represented as the product of prime numbers in a unique way.

• Prime Number: A number with exactly two factors: 1 and itself (e.g., 2, 3, 5, 7, 11).
• Composite Number: A number with more than two factors.

How the Calculator Finds Prime Factors

This tool uses a method often called a “Factor Tree.” It repeatedly divides your number by the smallest possible prime number until only 1 remains.

Example for the number 60:

1. 60 ÷ 2 = 30
2. 30 ÷ 2 = 15
3. 15 ÷ 3 = 5
4. 5 is prime.

Result: 2 x 2 x 3 x 5, or in exponential notation: 2^2 x 3 x 5.

Factoring Negative Numbers

Many calculators fail when you input a negative sign, but factors apply to negative integers as well. The rule is simple: if a x b = n, then -a x -b also equals n. However, when factoring a negative number (like -24), we look for pairs where one number is positive and the other is negative.

This calculator handles negatives by listing all integer combinations that result in your input. For Prime Factorization of negative numbers, we simply extract -1 as the first factor, leaving the rest as a standard positive prime decomposition.

Divisibility Rules

You don’t always need a calculator for small numbers.

• Divisible by 2: The number ends in 0, 2, 4, 6, or 8 (it is even).
• Divisible by 3: The sum of the digits is divisible by 3 (e.g., 123 → 1+2+3=6, so yes).
• Divisible by 4: The last two digits form a number divisible by 4 (e.g., 724).
• Divisible by 5: The number ends in 0 or 5.
• Divisible by 6: It is divisible by both 2 and 3.
• Divisible by 9: The sum of the digits is divisible by 9.
• Divisible by 10: The number ends in 0.

FAQs

Q1. Why is 1 not a prime number?

A: For a number to be prime, it must have exactly two distinct factors: 1 and itself. The number 1 only has one factor (1), so it fails the definition. It is a “unit,” neither prime nor composite.

Q2. What are the factors of 0?

A: Zero is unique. Every non-zero integer is a factor of 0 because any number multiplied by 0 equals 0. However, 0 cannot be a factor of any other number because division by zero is undefined.

Q3. What is the difference between Factoring and GCF?

A: Factoring is finding the parts of a single number. GCF (Greatest Common Factor) is finding the largest factor that two or more numbers share. You need to factor the numbers individually first to find their GCF.

Q4. Can this calculator factor polynomials?

A: No. This is an arithmetic factoring tool intended for integers. If you are looking to solve quadratic equations (like x^2 – 4), you need an algebraic solver. This tool is optimized for finding divisors of large integers, prime decomposition, and divisibility checks.

Sources: MathPapa, Symbolab, Calculator Soup, Mathway, WolframAlpha, Free Math Help, eMathHelp, Alpertron, Mathsite, Pearson.