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Difference between revisions of "2024 AMC 8 Problems/Problem 1"

(→‎Video by MathTalks_Now)
(→‎Solution 2)
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 = 1957532</cmath>
+
<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>.
  

Revision as of 05:20, 20 January 2025

Problem

What is the last digit of: \[222{,}222-22{,}222-2{,}222-222-22-2?\] $\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 8\qquad\textbf{(E) } 10$

Solution 1

We can rewrite the expression as $222,222-(22,222+2,222+222+22+2)$. We note that the units digit of $22,222+2,222+222+22+2$ is $0$ because all the units digits of the five numbers are $2$ and $5\cdot2=10$, which has a units digit of $0$. Now, we have something with a units digit of $0$ subtracted from $222,222$, and so the units digit of this expression is $\boxed{\textbf{(B) } 2}$.

Solution 2

\[222,222-22,222 = 200,000\] \[200,000 - 2,222 = 197778\] \[197778 - 222 = 197556\] \[197556 - 22 = 197534\] \[197534 - 2 = 197532\] So our answer is $\boxed{\textbf{(B) } 2}$.

Solution 3

We only care about the units digits. Thus, $2-2$ ends in $0$, $0-2$ after regrouping(10-2) ends in $8$, $8-2$ ends in $6$, $6-2$ ends in $4$, and $4-2$ ends in $\boxed{\textbf{(B) } 2}$.

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: \[(12-2)-(2+2+2+2)=10-8=\boxed{\textbf{(B) } 2}\]

Solution 5

\[222{,}222-22{,}222-2{,}222-222-22-2\equiv2-2-2-2-2\equiv-8\equiv\boxed{\textbf{(B) } 2}\pmod{10}\]


Solution 6 (easy!)

We can ignore the other digits and just do $22-2-2-2-2-2$. Because you are subtracting five $2s$ and $2*5 = 10$, you subtract $10$ from $22$. This gives us 12, so the last digit is $\boxed{\textbf{(B) } 2}$.

Video Solution by Central Valley Math Circle (Goes through full thought process)

https://youtu.be/-XcShDyuZIo

Video by MathTalks_Now

https://www.youtube.com/watch?v=crn37TRMLv4

-rc1219

Video Solution 1 (Detailed Explanation) 🚀⚡📊

Youtube Link ⬇️

https://youtu.be/jqsbMWhTYRg

~ ChillGuyDoesMath :)

Video Solution by Central Valley Math Circle (Goes through full thought process)

https://youtu.be/-XcShDyuZIo

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)

https://youtu.be/Ol1seWX0xHY

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

https://youtu.be/RzPadkHd3Yc

Video Solution 8 WhyMath

https://youtu.be/i4mcj3jRTxM

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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