Factor Calculator

What this calculator does

The Factor Calculator lists every positive whole number that divides a chosen number without a remainder. It also groups the factors into pairs and shows the prime factorization, which breaks the number into the prime numbers that multiply together to produce it.

Factors are useful when simplifying fractions, finding common denominators, studying divisibility, and solving number-theory problems. A prime number has exactly two positive factors: 1 and itself. A composite number has more than two factors.

How to use this calculator

1. Enter a positive whole number.
2. Click Calculate.
3. Review the complete factor list, the factor pairs, and the prime factorization.

The default example is 120. Its factors are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120. Its prime factorization is 120 = 2 × 2 × 2 × 3 × 5.

How factors are found

The calculator tests divisors from 1 up to the square root of the number. When a divisor divides the number evenly, both the divisor and the matching quotient are factors. This method finds every factor without testing every number up to the original value.

Factor pairs are written as (divisor, quotient). For 120, the first pair is (1, 120) and the last pair is (10, 12). Prime factors are found by dividing the number repeatedly by the smallest possible prime until only a prime number remains.

The relationship between factors and prime factors can be written as:

120 = 1 × 120
120 = 2 × 60
120 = 2 × 2 × 2 × 3 × 5

What the results mean

A factor pair shows two whole numbers whose product is the original number. The prime factorization shows the same number written as a product of primes. Every whole number greater than 1 has a unique prime factorization, apart from the order of the factors.

The calculator does not include negative factors. In algebra, negative factors are also valid, but the list would double in size and is usually not needed for arithmetic.

Assumptions and limitations

• The number must be a positive whole number from 1 to 1,000,000,000.
• The calculator lists positive factors only.
• The factor tree is not drawn, but the prime factorization contains the same information.
• For very large numbers, trial division can take longer; the upper limit keeps the calculation responsive.
• Zero is rejected because every number divides zero and the factor list would be infinite.

Frequently asked questions

Is 1 a prime number?

No. A prime number has exactly two distinct positive factors, and 1 has only one positive factor. The calculator therefore does not list 1 as a prime factor.

What is a factor pair?

A factor pair is two whole numbers that multiply to the original number, such as (8, 15) for 120. Reading the factor pairs from the outside inward is the same as reading the factor list from both ends.

How is the prime factorization calculated?

The calculator divides the number by 2 as many times as possible, then by 3, then by 5, and so on. Each successful division contributes a prime factor. When the remaining value is greater than 1, it is itself prime.

Can the calculator factor a negative number?

No. The calculator works with positive whole numbers. If you are factoring an algebraic expression and need negative factors, multiply the positive factors by −1.

Source

Reference: Calculator.net — Factor Calculator.