Difference between revisions of "2022 AMC 8 Problems/Problem 17"
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==Problem== | ==Problem== | ||
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If <math>n</math> is an even positive integer, the <math>\emph{double factorial}</math> 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 <math>\emph{double factorial}</math> 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> | ||
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~wamofan | ~wamofan | ||
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| + | ==Video Solution by Math-X (First understand the problem!!!)== | ||
| + | https://youtu.be/oUEa7AjMF2A?si=f4lLO32DQ4Yxdpkv&t=2925 | ||
| + | |||
| + | ~Math-X | ||
| + | |||
==Video Solution (🚀Under 2 min🚀)== | ==Video Solution (🚀Under 2 min🚀)== | ||
https://youtu.be/qOBzhBx8uw4 | https://youtu.be/qOBzhBx8uw4 | ||
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~savannahsolver | ~savannahsolver | ||
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| + | ==Video Solution 8== | ||
| + | https://www.youtube.com/watch?v=EVYrVkkpCo8 | ||
| + | |||
| + | ~Jamesmath | ||
==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}} | ||
Latest revision as of 01:04, 18 January 2024
Contents
Problem
If 
is an even positive integer, the 
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 
the units digit of will be 
because there will be a factor of 
Thus, we only need to calculate the units digit of ![\[2!!+4!!+6!!+8!! = 2+8+48+48\cdot8.\]](http://latex.artofproblemsolving.com/7/e/2/7e269223baea7ee3fc6200b9d06bb91f0c38b27e.png)
We only care about units digits, so we have 
which has the same units digit as 
The answer is 
~wamofan
Video Solution by Math-X (First understand the problem!!!)
https://youtu.be/oUEa7AjMF2A?si=f4lLO32DQ4Yxdpkv&t=2925
~Math-X
Video Solution (🚀Under 2 min🚀)
~Education, the Study of Everything
Video Solution
https://youtu.be/wp9tOyJ3YQY?t=146
Video Solution
https://youtu.be/Ij9pAy6tQSg?t=1461
~Interstigation
https://www.youtube.com/watch?v=FTVLuv_n9bY
~Ismail.Maths
Video Solution
https://youtu.be/hs6y4PWnoWg?t=80
~STEMbreezy
Video Solution
~savannahsolver
Video Solution 8
https://www.youtube.com/watch?v=EVYrVkkpCo8
~Jamesmath
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. 
