Factorials and Their Trailing Zeroes

This "trailing zeroes in a factorial" exercise is pretty easy to answer once you think about it the right way. I couldn't find anything much useful on the Internet, so here's a little lesson on how to handle it.

What are the trailing zeroes of a factorial?

If we take the factorial of any number larger than 5, then there will be at least one zero at the end of the number. Why? Because 5! = 1×2×3×4×5; in particular, 5! = (2×5)×(1×3×4), and (2×5) = 10. The factorial of any larger number will have more copies of 2 and 5 (as factors of larger values, like 6 and 15), so there will be even more factors of 10 in these factorials. And every factor of 10 adds a zero to the end of the factorial expansion.

Does every factorial have at least one zero at the end?

Not every factorial has at least one zero at the end of it. However, as soon as you get to 5!, which contains a factor of 2 and a factor of 5, you'll get a trailing zero. Larger factorials will have more zeroes. So the only factorials which do not have any trailing zeroes are 0!, 1!, 2!, and 4!.

This explains why there are trailing zeroes — because the factorials contain factors of 10 in them — but this doesn't tell us how to find the actual number of trailing zeroes. However, since the zeroes come from factors of 10, we can methodically figure out the actual number.

In the example below, I'll go through the reasoning which will then create a method for quickly answering this question.

• Find the number of trailing zeroes in the expansion of 23!

In fact, if I were to go to the trouble of multiplying out this factorial, I would be able to confirm that 23! = 25,852,016,738,884,976,640,000 does indeed have four trailing zeroes. I would also confirm that I really don't want to have to multiply things out; a logical method is going to be much better (that is, it is going to be much easier) than applying brute force.

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• Find the number of trailing zeroes in 101!

This reasoning, of finding the number of multiples of 5¹ = 5, plus the number of multiples of 5² = 25, etc, extends to working with even larger factorials.

• Find the number of trailing zeroes in the expansion of 1000!

The example above highlights the general method for answering this question, no matter what factorial they give you.

What are the steps for finding a factorial's trailing zeroes?

1. Take the number that you've been given the factorial of.
2. Divide by 5; if you get a decimal, truncate to a whole number.
3. Divide by 5² = 25; if you get a decimal, truncate to a whole number.
4. Divide by 5³ = 125; if you get a decimal, truncate to a whole number.
5. Continue with ever-higher powers of 5, until your division results in a number less than 1. Once the division is less than 1, stop.
6. Sum all the whole numbers that you got in your divisions. This is the number of trailing zeroes.

What do these steps look like, in application?

What is an example of finding the number of trailing zeroes in a factorial?

• How many trailing zeroes would be found in 4617!, upon expansion?

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By the way, you can get the same result, if you keep track as you go, by just dividing repeatedly in your calculator by 5's:

• 4617 ÷ 5 = 923.4 (write down 923)
• 923.4 ÷ 5 = 184.68 (write down 184)
• 184.68 ÷ 5 = 36.936 (write down 36)
• 36.936 ÷ 5 = 7.3827 (write down 7)
• 7.3827 ÷ 5 = 1.47744 (write down 1)
• 1.47744 ÷ 5 < 1 (so don't write anything down)

At which point, you're done doing divisions.

Turn to your scratch paper where you've written down the whole numbers (namely, 923,184, 36, 7, and 1), and add them up to get 1151, as before.

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Can software like Excel find the trailing zeroes for me?

In general, software like Excel won't help with this sort of computation, any more than your calculator could. Software customarily only stores fifteen or so digits of "accuracy", which is why, after a number gets sufficiently large, the display switches automatically to scientific notation.

Since the software is only storing the first few leading digits, the remaining trailing digits have to be filled in with zeroes. If you attempt the first expansion above, "23!", in Excel, you'll get something with way more trailing zeroes than is actually correct. In other words, the computer will give you the wrong answer.

So learn the concepts; don't try to cheat with software.