How do you find GCD by prime factorization?

Factor every number, then take the primes that appear in all inputs using the lowest exponent found for each prime.

How do you find LCM by prime factorization?

Factor every number, include every prime that appears, and use the highest exponent for each prime.

Why does 1 have no prime factors?

1 is the multiplicative identity, so its prime factorization is the empty product. The calculator displays it as 1.

Is this for very large integer factorization?

No. This browser tool is bounded to natural numbers up to 100,000,000 so the steps stay fast and explainable.

Prime Factorization Method Calculator

Factor multiple integers at once, compare prime exponents, and find the GCD and LCM from the minimum and maximum powers.

Numbers: at least 2 natural values, each up to 100,000,000

Use spaces, commas, semicolons, or line breaks. Example: 12 18 30 gives GCD 6 and LCM 180.

Show expanded prime factors

Prime factorization method for several numbers

English search intent for this page is less about the word canonical and more about using prime factorization to find GCD, LCM, or a common denominator. The table view keeps every prime exponent visible across all inputs.

n = p_{1}^{a_{1}} p_{2}^{a_{2}} \dots p_{k}^{a_{k}}

Each positive integer greater than 1 has a unique prime factorization.

Enter two or more natural numbers.
Read each number's prime factorization.
Use the minimum exponent in each prime column for the GCD.
Use the maximum exponent in each prime column for the LCM.

GCD and LCM from prime powers

Input	Prime factorization
12	x 3
18	2 x
30	2 x 3 x 5

g cd (12, 18, 30) = 2^{1} \cdot 3^{1} = 6

For the GCD, use only primes shared by every input and take the minimum exponent.

lcm (12, 18, 30) = 2^{2} \cdot 3^{2} \cdot 5 = 180

For the LCM, include every prime that appears and take the maximum exponent.

For a pure GCD problem with large numbers, the Euclidean algorithm is usually faster. This page is built for explainable prime-factorization work.

Frequently Asked Questions

Sources and References

Calculations are based on the listed reference sources. Links open in a new tab.

Updated: June 2, 2026

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