Prime Factorization Calculator

A division trail is more useful than a bare product

Enter 756 and the first successful divisor is 2. Dividing again leaves 189, then three divisions by 3 leave 7. The final prime factorization is 756 = 2² × 3³ × 7. The exponent records repetition, while the distinct-factor count is 3 because 2, 3 and 7 are counted once each.

What is a prime factor? It is a prime number that divides the original integer without a remainder. Prime factors are not an arbitrary decomposition: the Fundamental Theorem of Arithmetic states that every integer greater than 1 has one factorization, apart from the order in which the factors are written. Starting with 3 instead of 2 may change the working, but it cannot change the completed product.

The result also lists every positive divisor. For 72 = 2³ × 3², a divisor may use the exponent of 2 from 0 through 3 and the exponent of 3 from 0 through 2. That gives (3 + 1) × (2 + 1) = 12 divisors. Red labels mark the prime factors themselves; the other labels are products formed from those same powers.

• 72 becomes 2³ × 3² and has 12 positive divisors.
• 1,001 becomes 7 × 11 × 13 and has 8 positive divisors.
• 997 remains 997, so its only divisors are 1 and 997.

The square-root cutoff keeps the prime check finite

What is a prime number? Prime numbers are integers above 1 with exactly two positive divisors: 1 and the number itself. The smallest prime number is 2, which is also the only even prime. There is no largest prime number. A “largest known prime” is a computing record that can be replaced, not an endpoint to the sequence.

If an integer is composite, at least one member of a factor pair is no greater than its square root. Testing 997 therefore requires candidates only through roughly 31.6. Once none divides evenly, 997 is prime. A prime number table up to 100 helps with small candidates, beginning 2, 3, 5, 7, 11 and ending at 97. Absence from that table proves nothing about a larger value; 101 is prime even though the table stops before it.

The calculator performs this trial division for integers from 2 through 999,999,999. It rejects decimals, text mixed with digits and values outside that range. The step list is a checkable record of each exact division, while the copied result keeps the shorter exponent form, the distinct-factor count and the prime verdict.

Coprime describes a relationship between two integers

What does coprime mean? Two values are coprime when their greatest common divisor is 1. Coprime numbers do not have to be prime themselves: 8 and 9 are both composite, yet their factorizations 2³ and 3² share no factor. By contrast, 12 and 18 share 2 and 3.

A coprime calculator or coprime numbers checker needs two inputs. This page deliberately factorizes one integer at a time. You can compare two results by hand, but the GCD and LCM calculator is the direct route: a GCD of 1 confirms that the pair is coprime. Keeping these jobs separate also prevents a single-number prime test from being mistaken for a test of a pair.