How many zeros are there in 100!?
100! is 100 factorial, which is all the numbers up to 100 multiplied togther.
100! has 24 zeros.
Proof
The number of zeros at the end of a number tells us how many times 10 divides into the number. Each 10 that divides a number can be broken down into a 5 and a 2.
For factorial, factors of 2 are very common - every even number adds at least one 2. Thus for every 5, there will always be a 2 available. Therefore we only need to count how many times 5 goes into the factorial.
Every multiple of 5 introduces another 5 to the factorial’s factors. There are \(\left\lfloor \frac{n}{5} \right\rfloor\) 5s in the numbers up to \(n\). However, we also need to count an extra 5 for every multiple of \(5^2\), and so on for all powers of 5. We can only stop when the powers of 5 exceed \(n\).
Thus to calculate the number of times 5 divides \(n!\), and thus the number of zeros at the end, we have:
\[\sum_{i=1} \left\lfloor \frac{n}{5^i} \right\rfloor\]For \(n=100\):
\[\left\lfloor \frac{100}{5} \right\rfloor + \left\lfloor \frac{100}{25} \right\rfloor = 20+4 = 24\]