Factorial numbers, , grow very fast with . In fact, according to Stirling's approximation. The prime factors of a factorial number, however, are all relatively small, and the complete factorization of is quite easy to obtain.
We will make use of the following fundamental theorem:
for a prime , then or .
(Here, means that divides .) This is called Euclid's First Theorem or Euclid's Lemma. For most, it is intuitively clear, but a proof can be found in, e.g., Hardy and Wright: An Introduction to the Theory of Numbers.
An application of this theorem to factorial numbers is that if a prime is a divisor of then must be a divisor of at least one of the numbers . This immediately implies
Every prime factor of is less than or equal to .
Conversely, every prime number between 2 and must be a prime factor of .
Let us introduce the notation as the number of times divides into . Put more precisely, if and only if is an integer while is not.
We now seek to determine for all primes . From Euclid's First Theorem and the Fundamental Theorem of Arithmetic follows:
The trick here is not to consider the right-hand side term by term, but rather as a whole. Let us take
and as an example. How many of the numbers 1, 2, …, 42 are divisible by 3? Exactly of them. But this is not the total count, because some of them are divisible by 3 multiple times. So how many are divisible by ? of them. Similarly, . And . So we have
This procedure is easily generalized and we have
Doing this for all primes in our example, we get
Notice how the exponents do not increase as the prime numbers increase. This is true in general. Assume that and are both primes and . Then and for all positive integers . Using this in equation (1) we get
What about for composite numbers ? Given the factorization of both and , this is easy to compute. But if, e.g., the multiplicity of all prime factors of are the same, then the relation (2) can be used. Consider for a positive integer . Since then
But if then we can use (2) and we have
so there are 9 trailing zeros in the decimal representation of 42!.