Factor Calculator - Find Factors Step-by-Step

How to Find the Factors of a Number

Factors are whole numbers that divide a number exactly, leaving no remainder. Finding factors is useful for simplifying fractions, identifying common factors, working with divisibility, and solving problems in arithmetic and algebra.

Factors usually come in pairs. When two positive integers are multiplied together, their product is the original number. For example, 8 and 9 form a factor pair of 72 because 8 × 9 = 72.

How to Find Factors

1. Start with 1: Every positive integer greater than 0 has 1 as a positive factor.
2. Test divisibility: Check whole numbers to see which ones divide the number with no remainder.
3. Find the matching factor: Divide the original number by each factor to find its factor-pair partner.
4. Stop after completing the pairs: Once all factor pairs have been found, list the unique factors from smallest to largest.

Example: Find All Factors of 72

Let us find all the positive factors of 72 by looking for pairs of positive integers whose product is 72.

Step 1: Start with 1. Since 1 divides 72 exactly, its matching factor is 72.

Step 2: Check 2. Since 72 ÷ 2 = 36 with no remainder, 2 and 36 form another factor pair.

Step 3: Continue checking the remaining possible divisors. The other factor pairs are:

Step 4: Combine the numbers from each pair and arrange them in ascending order.

Therefore, 72 has 12 positive factors.

What Are Factor Pairs?

A factor pair consists of two integers whose product equals the original number. For 72, the positive factor pairs are:

Each number in these pairs is a factor of 72. Factor pairs make it easier to find and organize factors without listing the same factors repeatedly.

Prime Factors

A prime factor is a factor that is also a prime number. Prime factorization means expressing a number as a product of prime numbers.

For example, 72 can be divided by 2 repeatedly until all remaining factors are prime.

Combining the prime factors gives:

So the prime factors of 72 are 2 and 3, while its prime factorization is 2³ × 3².

Common Factors and Greatest Common Factor

A common factor is a factor shared by two or more numbers. The greatest common factor (GCF) is the largest positive factor that the numbers have in common.

For example, consider 18 and 24. Their positive factors include:

The common factors are 1, 2, 3, and 6. Therefore, the greatest common factor is:

Positive and Negative Factors

For a positive integer, factors are normally listed as positive factors. However, negative factors also exist because the product of two negative numbers is positive.

For example, both -2 and -36 are factors of 72 because:

If a problem asks for both positive and negative factors, each positive factor has a corresponding negative factor.

Factor Formula and Verification

A number is a factor of another number when division produces a whole-number result with no remainder. This can be written as:

You can also verify a factor by multiplying it by its matching factor. For example, 8 is a factor of 72 because 72 ÷ 8 = 9, and:

This multiplication check confirms that 8 and 9 form a valid factor pair.

Common Mistakes to Avoid

• Missing a factor pair: Check possible divisors systematically so that no valid pair is skipped.
• Including a non-factor: A factor must divide the original number exactly with no remainder.
• Listing duplicates: When using factor pairs, each factor should appear only once in the final list.
• Forgetting 1 and the number itself: Every positive integer greater than 0 has both as positive factors.
• Confusing factors with multiples: Factors divide a number evenly, while multiples are obtained by multiplying a number by whole numbers.
• Confusing prime factors with all factors: Prime factors are prime numbers that divide the original number; they are not necessarily the complete list of factors.

Why Use a Factor Calculator?

A Factor Calculator can quickly find the factors of an integer and help verify your work. Depending on the calculation, it can also help identify factor pairs, prime factors, or common factors, making it easier to check results without testing every possibility manually.