What is the units digit of the number $\sum_{n=1}^{1337}(n!)^4$

What can you say about the units digit of $n!$ for $n\ge5$?

What about the units digits for $n < 5$?

It’s sufficient to take only the units digit when multiplying or raising the number to any power.

All factorials greater that $5!$ have both the factors $5$ and $2$, hence the units digit equal to 0. This means that we only need to worry about $n \in \{1,2,3,4\}$. It’s sufficient to take only the units digit when multiplying or raising to any power.

We have:

• $1^4\rightarrow1$
• $(2!)^4=2^4\rightarrow6$
• $(3!)^4=6^4\rightarrow6$
• $(4!)^4=24^4\rightarrow4^4\rightarrow6$

The units digit for the sum is therefore the units digit of $1+6+6+6$, i.e. $9$.