Question

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:

$8!! = 2 \times 4 \times 6 \times 8$

What is the units digit of the following sum?

$2!! + 4!! + \cdots + 2022!!$

$0$

$2$

$4$

$6$

$8$

Solution:

If we take $n!!$ for an even $n$ that is greater or equal to $10,$ then $10$ is one of the numbers we multiply by. Since $10$ is a factor of $n!!,$ we know that the units digit of $10$ is $0,$ which means that it doesn't affect our result.

This means it suffices to compute the units digit of $2!!+4!!+6!!+8!!,$ which is equivalent to: $\begin{align*} &2+ 2(4)+2(4)(6)\\ &+2(4)(6)(8)\\ &= 2+8+48+384\\ &= 442 \end{align*}$ The units digit therefore is $2$

Thus, the correct answer is B.

2022 AMC 8 Problem 17

Solution:

Problem 17 in Other Years