**Legendre’s formula**:

The exponent of in the prime factorization of is

This is equivalent to:

Explanation: Between and there are numbers that are multiples of : . (e.g. if and , these are .) These each contribute a factor of . Of those, there are multiples of that each contribute one more factor of . (e.g. one more: . 4 factors in total.) The multiples of contribute one more factor, etc until .

Example: Find the exponent of in the prime factorization of .