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

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==Problem 1==
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==Problem==
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>
 
<math>\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8</math>
 
<math>\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8</math>
  
 
==Solution 1==
 
==Solution 1==
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>.
 
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>.
 
 
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>
<math>222,222-22,222 = 200,000</math>
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<cmath>200,000 - 2,222 = 197778</cmath>
<math>200,000 - 2,222 = 197778</math>
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<cmath>197778 - 222 = 197556</cmath>
<math>197778 - 222 = 197556</math>
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<cmath>197556 - 22 = 197534</cmath>
<math>197556 - 22 = 197534</math>
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<cmath>197534 - 2 = 1957532</cmath>
<math>197534 - 2 = 1957532
 
</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>.
  
We only care about the unit's digits.
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==Solution 4==
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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>
  
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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== 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>
  
-unknown
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== Video Solution 1 (Detailed Explanation)==
  
minor edits by Fireball9746
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https://youtu.be/jqsbMWhTYRg
  
==Solution 4==
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==Video Solution by Central Valley Math Circle (Goes through full thought process)==
 +
https://youtu.be/-XcShDyuZIo
  
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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==Video Solution 2 (MATH-X)==
<cmath>(12-2)-(2+2+2+2)=10-8=2</cmath>
 
Thus, we get the answer <math>\boxed{(B)}</math>
 
 
 
==Video Solution (MATH-X)==
 
 
https://youtu.be/BaE00H2SHQM?si=O0O0g7qq9AbhQN9I&t=130
 
https://youtu.be/BaE00H2SHQM?si=O0O0g7qq9AbhQN9I&t=130
  
~Math-X
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==Video Solution 3 (A Clever Explanation You’ll Get Instantly)==
 
 
==Video Solution (A Clever Explanation You’ll Get Instantly)==
 
 
https://youtu.be/5ZIFnqymdDQ?si=IbHepN2ytt7N23pl&t=53
 
https://youtu.be/5ZIFnqymdDQ?si=IbHepN2ytt7N23pl&t=53
  
~hsnacademy
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==Video Solution  4 (Quick and Easy)==
 
 
==Video Solution  (Quick and Easy!)==
 
 
https://youtu.be/Ol1seWX0xHY
 
https://youtu.be/Ol1seWX0xHY
  
~Education, the Study of Everything
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==Video Solution 5 Interstigation==
 
 
==Video Solution by Interstigation==
 
 
https://youtu.be/ktzijuZtDas&t=36
 
https://youtu.be/ktzijuZtDas&t=36
  
==Video Solution by Daily Dose of Math==
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==Video Solution 6 Daily Dose of Math==
 
 
 
https://youtu.be/bSPWqeNO11M?si=HIzlxPjMfvGM5lxR
 
https://youtu.be/bSPWqeNO11M?si=HIzlxPjMfvGM5lxR
  
~Thesmartgreekmathdude
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==Video Solution 7 Dr. David==
 
 
==Video Solution by Dr. David==
 
 
 
 
https://youtu.be/RzPadkHd3Yc
 
https://youtu.be/RzPadkHd3Yc
  
==Video Solution by WhyMath==
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==Video Solution 8 WhyMath==
 
https://youtu.be/i4mcj3jRTxM
 
https://youtu.be/i4mcj3jRTxM
  

Latest revision as of 19:58, 13 January 2025

Problem

What is the last digit of: \[222{,}222-22{,}222-2{,}222-222-22-2?\] $\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8$

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

\[222,222-22,222 = 200,000\] \[200,000 - 2,222 = 197778\] \[197778 - 222 = 197556\] \[197556 - 22 = 197534\] \[197534 - 2 = 1957532\] 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

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

Video Solution 1 (Detailed Explanation)

https://youtu.be/jqsbMWhTYRg

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 (ProblemsAnswer KeyResources)
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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