The **fundamental theorem of arithmetic** is the statement that every positive integer has a unique prime number factorization. For instance, we can write
*a* ≤ 3, 0 ≤ *b* ≤ 1, and 0 ≤ *c* ≤ 2. This yields a total of 4 × 2 × 3 = 24 positive factors.

- 6936 = 2
^{3}× 3 × 17^{2}

Knowing the prime number factorization of a number gives complete knowledge about all factors of that number. For instance, the above factorization of 6936 tells us that the positive factors of 6936 are of the form

- 2
^{a}× 3^{b}× 17^{c}

Once the prime factorizations of two numbers are known, their greatest common divisor and least common multiple can be found quickly. However if the prime factorizations are not known, the use of Euclid's algorithm is likely to require less calculation than factorizing the two numbers.

The fundamental theorem ensures that multiplicative functions are completely determined by their values on the powers of prime numbers.