Zeros at the End of a Factorial


How many trailing zeros does \(n!\) have?


Consider a fairly small factorial, like \(10! = 3,628,800\), which has \(2\) trailing zeros. Is there an easy way to see why it has \(2\) trailing zeros without performing the multiplications directly?

\[10! = 10\times9\times8\times7\times6\times5\times4\times3\times2\times1\]

Looking at the above, we can see why \(10!\) has \(2\) trailing zeros. Obviously, the factor \(10\) contributes \(1\) of these. The other trailing zero comes from the product of the factors \(5\) and \(2\), which also gives \(10\).

This idea of counting factors that are multiples of \(5\) is a good initial approach. Among these, multiples of \(10\) immediately contribute a trailing zero, while the remaining multiples of \(5\) contribute a trailing zero when multiplied by an even factor. Since half of the numbers in the factorial’s expansion are even, there is always an even number to pair with a multiple of \(5\).

Let’s apply this approach to \(15!\): \[15! = 15\times14\times13\times12\times11\times 10\times9\times8\times7\times6\times5\times4\times3\times2\times1\]

This factorial contains \(3\) multiples of \(5\) (\(5\), \(10\), and \(15\)), which gives a total of \(3\) trailing zeros. Performing the multiplications gives \(15! = 1,307,674,368,000\), which confirms the solution.

Now, let’s try applying the approach to \(25!\). We know the expansion of \(25!\) contains \(5\) multiples of \(5\), which are \(5\), \(10\), \(15\), \(20\), and \(25\). Thus, applying our method gives \(5\) trailing zeros. In reality, however, \(25!\) has \(6\) trailing zeros: \[25!= 15,511,210,043,330,985,984,000,000\]

What gives? The issue is that one term in this factorial’s expansion is \(25\) (or \(5^2\)), which contains two \(5\)s. Thus, it actually contributes \(2\) trailing zeros to the count instead of just \(1\). This shows us that we must account for powers of \(5\) specially. For example, \(5^3=125\) and \(5^4=625\) contribute \(3\) and \(4\) trailing zeros, respectively.

Fortunately, we can easily represent this process mathematically. If we define \(z_n\) as the number of trailing zeros for \(n!\), we can write the formula as follows:

\[z_n = \left\lfloor\frac{n}{5}\right\rfloor+ \left\lfloor\frac{n}{5^2}\right\rfloor+ \left\lfloor\frac{n}{5^3}\right\rfloor+ \left\lfloor\frac{n}{5^4}\right\rfloor+ \ldots\]

The above formula only needs to be extended to the largest product of \(5\) that is less than or equal to \(n\). For example, for \(125!\) we extend it to the \(3\)3rd term, which gives:

\[z_{125} = \left\lfloor\frac{125}{5}\right\rfloor+ \left\lfloor\frac{125}{5^2}\right\rfloor+ \left\lfloor\frac{125}{5^3}\right\rfloor= 25 + 5 + 1 = 31\]