Factorials & Trailing Zeros

At a glance 11 formulas

1. 1Factorial$n! = 1 \times 2 \times 3 \times \cdots \times n$$0! = 1,\quad (n+1)! = (n+1) \times n!$
2. 2Values to know

   $4!$

   24

   $5!$

   120

   $6!$

   720

   $7!$

   5,040

   $8!$

   40,320

   $9!$

   3,62,880

   $10!$

   36,28,800

3. 3Factor out the smallestTake the smallest factorial out as a common factor: $7! + 8! + 9! = 7!(1 + 8 + 72) = 81 \times 7!$.
4. 4Prime power in n!$\left\lfloor \tfrac{n}{p} \right\rfloor + \left\lfloor \tfrac{n}{p^{2}} \right\rfloor + \left\lfloor \tfrac{n}{p^{3}} \right\rfloor + \cdots$
5. 5Divide and addDivide $n$ by $p$, then the quotient by $p$, and so on, and add the quotients. For 2 in $50!$: $25 + 12 + 6 + 3 + 1 = 47$.
6. 6Trailing zeros$Z(n!) = \left\lfloor \tfrac{n}{5} \right\rfloor + \left\lfloor \tfrac{n}{25} \right\rfloor + \left\lfloor \tfrac{n}{125} \right\rfloor + \cdots$$\text{2s always outnumber 5s in } n!$
7. 7Power of a compositeSplit it into primes, count each prime in $n!$, divide by its exponent and take the smallest: for $18 = 2 \times 3^{2}$, use $\min(e_2, \lfloor e_3/2 \rfloor)$.
8. 8Zeros in any productCount the 2s and the 5s in the whole product; the number of zeros at the end is the smaller count.
9. 9Zero counts that skipEach multiple of 25 adds two or more zeros at once, so some counts never occur: no $n!$ ends in exactly 5 or 11 zeros. Any count above 0 that occurs fits exactly 5 values of $n$.
10. 10Remainders of factorial sumsOnce $n!$ is a multiple of the divisor, so is every later term: by 15, $1! + 2! + \cdots + 40!$ leaves what $1 + 2 + 6 + 24 = 33$ leaves, 3.
11. 11Wilson's theorem$(p-1)! \equiv -1 \pmod p$$(p-2)! \equiv 1 \pmod p$