2022 AMC 8 Problems/Problem 17

Problem

If $n$ is an even positive integer, the double factorial notation $n!!$ represents the product of all the even integers from $2$ to $n$ . For example, $8!! = 2 \cdot 4 \cdot 6 \cdot 8$ . What is the units digit of the following sum? $2!! + 4!! + 6!! + \cdots + 2018!! + 2020!! + 2022!!$

$\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8$

Solution

Notice that once $n>8,$ $n!!$ ’s units digit will be $0$ because there will be a factor of $10$ . Thus, we only need to calculate the units digit of $2!!+4!!+6!!+8!! = 2+2\cdot 4+2\cdot 4\cdot 6 + 2\cdot 4\cdot 6\cdot 8 = 2+8+48+48\cdot 8.$ We only care about units digits so we have $2+8+8+8\cdot 8$ Which is $2+8+8+4= \boxed{\textbf{(B) }2}$

~wamofan

2022 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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2022 AMC 8 Problems/Problem 17

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