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

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==Problem 1==
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==Problem==
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What is the ones digit of: <cmath>222{,}222-22{,}222-2{,}222-222-22-2?</cmath>
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What is the last digit of: <cmath>222{,}222-22{,}222-2{,}222-222-22-2?</cmath>
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<math>\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8</math>
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<math>\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 8\qquad\textbf{(E) } 10</math>
  
 
==Solution 1==
 
==Solution 1==
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We can rewrite the expression as <cmath>222,222-(22,222+2,222+222+22+2).</cmath>
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We can rewrite the expression as <math>222,222-(22,222+2,222+222+22+2)</math>. We note that the units digit of <math>22,222+2,222+222+22+2</math> is <math>0</math> because all the units digits of the five numbers are <math>2</math> and <math>5\cdot2=10</math>, which has a units digit of <math>0</math>. Now, we have something with a units digit of <math>0</math> subtracted from <math>222,222</math>, and so the units digit of this expression is <math>\boxed{\textbf{(B) } 2}</math>.
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We note that the units digit of the addition is <math>0</math> because all the units digits of the five numbers are <math>2</math> and <math>5*2=10</math>, which has a units digit of <math>0</math>.
 
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Now, we have something with a units digit of <math>0</math> subtracted from <math>222,222</math>. The units digit of this expression is obviously <math>2</math>, and we get <math>\boxed{B}</math> as our answer.
 
  
 
==Solution 2==
 
==Solution 2==
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<cmath>222,222-22,222 = 200,000</cmath>
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<math>222,222-22,222 = 200,000</math>
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<cmath>200,000 - 2,222 = 197778</cmath>
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<math>200,000 - 2,222 = 197778</math>
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<cmath>197778 - 222 = 197556</cmath>
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<math>197778 - 222 = 197556</math>
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<cmath>197556 - 22 = 197534</cmath>
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<math>197556 - 22 = 197534</math>
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<cmath>197534 - 2 = 197532</cmath>
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<math>197534 - 2 = 1957532
 
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</math>
 
 
So our answer is <math>\boxed{\textbf{(B) } 2}</math>.
 
So our answer is <math>\boxed{\textbf{(B) } 2}</math>.
  
 
==Solution 3==
 
==Solution 3==
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We only care about the units digits. Thus, <math>2-2</math> ends in <math>0</math>, <math>0-2</math> after regrouping(10-2) ends in <math>8</math>, <math>8-2</math> ends in <math>6</math>, <math>6-2</math> ends in <math>4</math>, and <math>4-2</math> ends in <math>\boxed{\textbf{(B) } 2}</math>.
 +
 +
==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:
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<cmath>(12-2)-(2+2+2+2)=10-8=\boxed{\textbf{(B) } 2}</cmath>
  
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We only care about the unit's digits.
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== Solution 5 ==
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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>
  
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Thus, <math>2-2</math> ends in <math>0</math>, <math>0-2</math> after regrouping(10-2) ends in <math>8</math>, <math>8-2</math> ends in <math>6</math>, <math>6-2</math> ends in <math>4</math>, and <math>4-2</math> ends in  <math>\boxed{\textbf{(B) } 2}</math>.
 
  
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-unknown
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== Solution 6==
  
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minor edits by Fireball9746
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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>.
  
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==Solution 4==
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== Video Solution 1 (Detailed Explanation) 🚀⚡📊 ==
 +
Youtube Link ⬇️
 +
 
 +
https://youtu.be/jqsbMWhTYRg
  
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We just take the units digit of each and subtract, or you can do it this way by adding an extra ten to the first number (so we don't get a negative number):
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~ ChillGuyDoesMath :)
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<cmath>(12-2)-(2+2+2+2)=10-8=2</cmath>
 
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Thus, we get the answer <math>\boxed{(B)}</math>
 
  
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==Video Solution (A Clever Explanation You’ll Get Instantly)==
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== Video by MathTalks_Now ==
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https://youtu.be/5ZIFnqymdDQ?si=IbHepN2ytt7N23pl&t=53
 
  
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~hsnacademy
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https://www.youtube.com/watch?v=crn37TRMLv4
  
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==Video Solution 1 (Quick and Easy!)==
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-rc1219
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https://youtu.be/Ol1seWX0xHY
 
  
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~Education, the Study of Everything
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==Video Solution by Central Valley Math Circle (Goes through full thought process)==
 +
https://youtu.be/-XcShDyuZIo
  
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==Video Solution (easy to understand)==
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==Video Solution 2 (MATH-X)==
 
https://youtu.be/BaE00H2SHQM?si=O0O0g7qq9AbhQN9I&t=130
 
https://youtu.be/BaE00H2SHQM?si=O0O0g7qq9AbhQN9I&t=130
  
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~Math-X
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==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
  
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==Video Solution by Interstigation==
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==Video Solution 5 Interstigation==
 
https://youtu.be/ktzijuZtDas&t=36
 
https://youtu.be/ktzijuZtDas&t=36
  
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==Video Solution by Daily Dose of Math==
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==Video Solution 6 Daily Dose of Math==
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https://youtu.be/bSPWqeNO11M?si=HIzlxPjMfvGM5lxR
 
https://youtu.be/bSPWqeNO11M?si=HIzlxPjMfvGM5lxR
  
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~Thesmartgreekmathdude
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==Video Solution 7 Dr. David==
 +
https://youtu.be/RzPadkHd3Yc
  
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==Video Solution by Dr. David==
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==Video Solution 8 WhyMath==
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https://youtu.be/i4mcj3jRTxM
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https://youtu.be/RzPadkHd3Yc
 
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2024|before=First Problem|num-a=2}}
 
{{AMC8 box|year=2024|before=First Problem|num-a=2}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 08:46, 16 February 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

We can ignore the other digits and just do $22-2-2-2-2-2$. Because you are subtracting five $2s$ and $2\cdot5 = 10$, you subtract $10$ from $22$. This gives us 12, so the last digit is $\boxed{\textbf{(B) } 2}$.

Video Solution 1 (Detailed Explanation) 🚀⚡📊

Youtube Link ⬇️

https://youtu.be/jqsbMWhTYRg

~ 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)

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

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