Factor Calculator

Formula

n = p₁ᵃ × p₂ᵇ × ...

Any composite number can be written as a unique product of prime numbers.

Worked Examples

Example 1: All Factors of 12 by Pairing

Problem:List every factor of 12.

Solution:Test each integer from 1 up to the square root of 12 (about 3.46) and record both members of each pair. 1 × 12 2 × 6 3 × 4 4 is already listed as the partner of 3, so the search stops. Factors: 1, 2, 3, 4, 6, 12 - six of them. Prime factorisation 12 = 2² × 3 predicts that count: (2+1)(1+1) = 6.

Result:1, 2, 3, 4, 6, 12 (six factors)

Example 2: Prime Factorisation of 100

Problem:Write 100 as a product of primes and count its factors.

Solution:Divide repeatedly by the smallest prime that fits. 100 / 2 = 50 50 / 2 = 25 25 / 5 = 5 5 / 5 = 1 Prime factorisation: 100 = 2² × 5² Number of factors = (2+1)(2+1) = 9: they are 1, 2, 4, 5, 10, 20, 25, 50, 100. Because every exponent is even, 100 is a perfect square, with the square root 2 × 5 = 10.

Result:100 = 2² × 5² | 9 factors | perfect square (10²)

Example 3: Factor Count and Divisor Sum for 36

Problem:Find the factors of 36, how many there are, and what they add up to.

Solution:Prime factorisation: 36 = 2² × 3² Factor count = (2+1)(2+1) = 9 Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36 Sum of factors = (1+2+4)(1+3+9) = 7 × 13 = 91 Direct check: 1+2+3+4+6+9+12+18+36 = 91. Matches. 36 has an odd number of factors, which happens only for perfect squares, because 6 pairs with itself.

Result:9 factors, summing to 91 | 36 = 2² × 3² = 6²

Frequently Asked Questions

What is a perfect number?

A number whose factors (excluding itself) sum up to the number. Example: 6 (1+2+3=6).

Why is prime factorization useful?

It helps in finding GCF and LCM, simplifying fractions, and is the basis of modern cryptography (RSA encryption).

Can a number have an odd number of factors?

Yes, only perfect squares (like 9, 16, 25) have an odd number of factors.

How do I check if a large number is prime?

You can use trial division up to the square root of the number. If no factors are found by then, it is prime.

What are twin primes?

Twin primes are pairs of prime numbers that differ by 2, such as (3, 5), (11, 13), and (17, 19).

What is the fundamental theorem of arithmetic?

It states that every integer greater than 1 is either a prime or can be represented as the product of prime numbers in a unique way.

Background & Theory

Factors:
Factors are integers that divide a number evenly (result is a whole number).

Example: Factors of 10 are 1, 2, 5, 10.

Prime Factorization:
Breaking a number down into the set of prime numbers which multiply together to result in the original number.

Example: 12 = 2 × 2 × 3
This "fingerprint" is unique for every number.

Prime Numbers:
Numbers that have exactly two factors: 1 and themselves (e.g., 2, 3, 5, 7, 11).

Composite Numbers:
Numbers that have more than two factors.

History

The study of factors and prime numbers is a cornerstone of number theory, dating back to ancient Greece. Euclid (c. 300 BC) proved the fundamental theorem of arithmetic, which states that every integer greater than 1 is either a prime itself or can be represented as the product of prime numbers in a unique way. Eratosthenes (c. 276-194 BC) created the "Sieve of Eratosthenes," a simple algorithm for finding all prime numbers up to a specified integer, which is still taught in schools today.