2024 AMC 8 Problems/Problem 1: Difference between revisions
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==Problem== | ==Problem== | ||
What is the | What is the unit digit of: <cmath>222{,}222-22{,}222-2{,}222-222-22-2?</cmath> | ||
<math>\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 8\qquad\textbf{(E) } 10</math> | <math>\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 8\qquad\textbf{(E) } 10</math> | ||
| Line 11: | Line 11: | ||
<cmath>197778 - 222 = 197556</cmath> | <cmath>197778 - 222 = 197556</cmath> | ||
<cmath>197556 - 22 = 197534</cmath> | <cmath>197556 - 22 = 197534</cmath> | ||
<cmath>197534 - 2 = | <cmath>197534 - 2 = 197532</cmath> | ||
So our answer is <math>\boxed{\textbf{(B) } 2}</math>. | So our answer is <math>\boxed{\textbf{(B) } 2}</math>. (Note that brute forcing takes lots of time and the AMC8 is a timed test) | ||
==Solution 3== | ==Solution 3== | ||
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<cmath>(12-2)-(2+2+2+2)=10-8=\boxed{\textbf{(B) } 2}</cmath> | <cmath>(12-2)-(2+2+2+2)=10-8=\boxed{\textbf{(B) } 2}</cmath> | ||
== | == Solution 5 == | ||
<cmath> \boxed{\textbf{ | <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 | == Solution 6== | ||
= | 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\cdot5 = 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 Solution 1 (Detailed Explanation) 🚀⚡📊 == | |||
Youtube Link ⬇️ | Youtube Link ⬇️ | ||
| Line 36: | Line 35: | ||
~ ChillGuyDoesMath :) | ~ ChillGuyDoesMath :) | ||
== Video by MathTalks_Now == | |||
https://www.youtube.com/watch?v=crn37TRMLv4 | |||
-rc1219 | |||
==Video Solution by Central Valley Math Circle (Goes through full thought process)== | ==Video Solution by Central Valley Math Circle (Goes through full thought process)== | ||
| Line 60: | Line 65: | ||
==Video Solution 8 WhyMath== | ==Video Solution 8 WhyMath== | ||
https://youtu.be/i4mcj3jRTxM | https://youtu.be/i4mcj3jRTxM | ||
== Video solution by TheNeuralMathAcademy == | |||
https://youtu.be/f63MY1T2MgI&t=0s | |||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2024| | {{AMC8 box|year=2024|num-b=First Problem|num-a=2}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
[[Category:Introductory Combinatorics Problems]] | |||
Latest revision as of 20:53, 20 August 2025
Problem
What is the unit 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)
![\[197534 - 2 = 197532\]](http://latex.artofproblemsolving.com/8/e/e/8ee577692ffa6b0660b721ceb15519c8e60b3297.png)
So our answer is . (Note that brute forcing takes lots of time and the AMC8 is a timed test)
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
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 Solution 1 (Detailed Explanation) 🚀⚡📊
Youtube Link ⬇️
~ ChillGuyDoesMath :)
Video by MathTalks_Now
https://www.youtube.com/watch?v=crn37TRMLv4
-rc1219
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
Video solution by TheNeuralMathAcademy
https://youtu.be/f63MY1T2MgI&t=0s
See Also
| 2024 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 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
