What is the units digit of the number
$\sum_{n=1}^{1337}(n!)^4$
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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.
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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.$9$.