Difference between revisions of "2022 AMC 8 Problems/Problem 17"
m (→Problem) |
(Added solution) |
||
| Line 3: | Line 3: | ||
If <math>n</math> is an even positive integer, the <i>double factorial</i> notation <math>n!!</math> represents the product of all the even integers from <math>2</math> to <math>n</math>. For example, <math>8!! = 2 \cdot 4 \cdot 6 \cdot 8</math>. What is the units digit of the following sum? <cmath>2!! + 4!! + 6!! + \cdots + 2018!! + 2020!! + 2022!!</cmath> | If <math>n</math> is an even positive integer, the <i>double factorial</i> notation <math>n!!</math> represents the product of all the even integers from <math>2</math> to <math>n</math>. For example, <math>8!! = 2 \cdot 4 \cdot 6 \cdot 8</math>. What is the units digit of the following sum? <cmath>2!! + 4!! + 6!! + \cdots + 2018!! + 2020!! + 2022!!</cmath> | ||
| − | <math>\textbf{(A)} | + | <math>\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8\qquad</math> |
==Solution== | ==Solution== | ||
| + | |||
| + | Notice that once <math>n>8,</math> <math>n!!</math>’s units digit will be <math>0</math> because there will be a factor of <math>10</math>. Thus, we only need to calculate the units digit of <math>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.</math> We only care about units digits so we have <math>2+8+8+8\cdot 8</math> Which is <math>2+8+8+4= \boxed{\textbf{(B) }2}</math> | ||
| + | |||
| + | ~wamofan | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2022|num-b=16|num-a=18}} | {{AMC8 box|year=2022|num-b=16|num-a=18}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 13:11, 28 January 2022
Problem
If 
is an even positive integer, the double factorial notation 
represents the product of all the even integers from 
to . For example, 
. What is the units digit of the following sum? ![\[2!! + 4!! + 6!! + \cdots + 2018!! + 2020!! + 2022!!\]](http://latex.artofproblemsolving.com/6/b/7/6b7552c1c8d367a79bf701d4631efea748362631.png)
Solution
Notice that once 
’s units digit will be 
because there will be a factor of 
. Thus, we only need to calculate the units digit of 
We only care about units digits so we have 
Which is 
~wamofan
See Also
| 2022 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 16 |
Followed by Problem 18 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
