2024 AMC 8 Problems/Problem 1: Difference between revisions
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<cmath>222{,}222-22{,}222-2{,}222-222-22-2\equiv2-2-2-2-2\equiv-8\equiv\boxed{\textbf{(B) } 2}\pmod{10}</cmath> | <cmath>222{,}222-22{,}222-2{,}222-222-22-2\equiv2-2-2-2-2\equiv-8\equiv\boxed{\textbf{(B) } 2}\pmod{10}</cmath> | ||
== Solution 6 (easy!)== | |||
We can ignore the other digits and just do <math>22-2-2-2-2-2</math>. Because you are subtracting five <math>2s</math> and <math>2*5 = 10</math>, you subtract <math>10</math> from <math>22</math>. This gives us 12, so the last digit is <math>\boxed{\textbf{(B) } 2}</math>. | |||
== Video by MathTalks_Now == | == Video by MathTalks_Now == | ||
Revision as of 17:12, 19 January 2025
Problem
What is the last digit of: ![\[222{,}222-22{,}222-2{,}222-222-22-2?\]](http://latex.artofproblemsolving.com/1/5/6/156c3662fe4c874a80e54887f76c302c13b9d14c.png)
Solution 1
We can rewrite the expression as 
. We note that the units digit of 
is 
because all the units digits of the five numbers are 
and 
, which has a units digit of . Now, we have something with a units digit of subtracted from 
, and so the units digit of this expression is 
.
Solution 2
![\[222,222-22,222 = 200,000\]](http://latex.artofproblemsolving.com/6/5/f/65f267d69bed2998343ebacfb03bdb51895befe1.png)
![\[200,000 - 2,222 = 197778\]](http://latex.artofproblemsolving.com/e/0/f/e0f3feeaabb6a0475cbc4a4e1e671405be4bfd9a.png)
![\[197778 - 222 = 197556\]](http://latex.artofproblemsolving.com/c/b/6/cb673ea3ee34766e705954c7d27eaf8c67d46c7e.png)
![\[197556 - 22 = 197534\]](http://latex.artofproblemsolving.com/a/c/9/ac90567551c4edf02afb197258e816e4a85f05ca.png)
![\[197534 - 2 = 1957532\]](http://latex.artofproblemsolving.com/e/a/9/ea985f5c7c1dcb748bec63329a9c8df894121820.png)
So our answer is .
Solution 3
We only care about the units digits. Thus, 
ends in , 
after regrouping(10-2) ends in 
, 
ends in 
, 
ends in 
, and 
ends in .
Solution 4
We just take the units digit of each and subtract, adding an extra ten to the first number so we don't get a negative number:
![\[(12-2)-(2+2+2+2)=10-8=\boxed{\textbf{(B) } 2}\]](http://latex.artofproblemsolving.com/a/9/f/a9f31cf010b9cbb2639ce1593ce0eab12ef1294c.png)
Solution 5
![\[222{,}222-22{,}222-2{,}222-222-22-2\equiv2-2-2-2-2\equiv-8\equiv\boxed{\textbf{(B) } 2}\pmod{10}\]](http://latex.artofproblemsolving.com/d/1/1/d117de6eed4b55e6a27b81ab3c47300579334b87.png)
Solution 6 (easy!)
We can ignore the other digits and just do 
. Because you are subtracting five 
and 
, you subtract 
from 
. This gives us 12, so the last digit is .
Video by MathTalks_Now
https://www.youtube.com/watch?v=crn37TRMLv4
-rc1219
Video Solution 1 (Detailed Explanation) 🚀⚡📊
Youtube Link ⬇️
~ ChillGuyDoesMath :)
Video Solution by Central Valley Math Circle (Goes through full thought process)
Video Solution 2 (MATH-X)
https://youtu.be/BaE00H2SHQM?si=O0O0g7qq9AbhQN9I&t=130
Video Solution 3 (A Clever Explanation You’ll Get Instantly)
https://youtu.be/5ZIFnqymdDQ?si=IbHepN2ytt7N23pl&t=53
Video Solution 4 (Quick and Easy)
Video Solution 5 Interstigation
https://youtu.be/ktzijuZtDas&t=36
Video Solution 6 Daily Dose of Math
https://youtu.be/bSPWqeNO11M?si=HIzlxPjMfvGM5lxR
Video Solution 7 Dr. David
Video Solution 8 WhyMath
See Also
| 2024 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by First Problem |
Followed by Problem 2 | |
| 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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing
