Math CentralQuandaries & Queries


Question from Alicia, a student:

What is the tens digit of 0! + 1! + 2! + 3! + ... + 2000!?


Write down a few factorials

0! = 1
1! = 1
2! = 2
3! = 6

When is the first time you get a units digit of 0? For every larger integer the units digit of its factorial is also 0. Why?

When is the first time you get a units digit and tens digit both 0? What about the factorials of larger integers?

I hope this helps,



(1) Compute the first several factorials. Which is the first one divisible by 10? By 100?

(2) Show that every factorial after that one is also divisible by 100.

(3) Which terms in the sum affect the tens digit nontrivially?

(4) The answer now follows from a simple calculation.

Victory lap: find the hundreds digit of the same sum! It will take you very little more time.

Good Hunting!

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