Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

(→‎Video Solution 1 (Detailed Explanation) 🚀⚡📊)
m (→‎Solution 5: Fix LaTeX)
 
(3 intermediate revisions by 3 users not shown)
Line 1: Line 1:
−
==Problem 1==
+
==Problem==
 
What is the ones digit of: <cmath>222{,}222-22{,}222-2{,}222-222-22-2?</cmath>
 
What is the ones 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>
+
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==
−
 
+
<cmath>222,222-22,222 = 200,000</cmath>
−
<math>222,222-22,222 = 200,000</math>
+
<cmath>200,000 - 2,222 = 197778</cmath>
−
<math>200,000 - 2,222 = 197778</math>
+
<cmath>197778 - 222 = 197556</cmath>
−
<math>197778 - 222 = 197556</math>
+
<cmath>197556 - 22 = 197534</cmath>
−
<math>197556 - 22 = 197534</math>
+
<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==
−
 
+
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.
 
−
 
 
−
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>.
 
−
 
 
−
-unknown
 
−
 
 
−
minor edits by Fireball9746
 
  
 
==Solution 4==
 
==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:
 +
<cmath>(12-2)-(2+2+2+2)=10-8=\boxed{\textbf{(B) } 2}</cmath>
  
−
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):
+
== Solution 5 ==
−
<cmath>(12-2)-(2+2+2+2)=10-8=2</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>
−
Thus, we get the answer <math>\boxed{(B)}</math>
 
  
−
== Video Solution 1 (Detailed Explanation) 🚀⚡📊 ==
+
==Video Solution by Central Valley Math Circle (Goes through full thought process)==
 +
https://youtu.be/-XcShDyuZIo
  
−
Youtube Link ⬇️
+
~mr_mathman
  
−
bit.ly/2024AMC8Problem1Solution
+
== Video Solution 1 (Detailed Explanation) ==
 +
https://youtu.be/jqsbMWhTYRg
  
 
~ ChillGuyDoesMath :)
 
~ ChillGuyDoesMath :)
  
−
==Video Solution (MATH-X)==
+
==Video Solution 2 (MATH-X)==
 
https://youtu.be/BaE00H2SHQM?si=O0O0g7qq9AbhQN9I&t=130
 
https://youtu.be/BaE00H2SHQM?si=O0O0g7qq9AbhQN9I&t=130
  
−
~Math-X
+
==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
 
~hsnacademy
  
−
==Video Solution  (Quick and Easy!)==
+
==Video Solution  4 (Quick and Easy!)==
 
https://youtu.be/Ol1seWX0xHY
 
https://youtu.be/Ol1seWX0xHY
  
Line 63: Line 51:
  
 
==Video Solution by Daily Dose of Math==
 
==Video Solution by Daily Dose of Math==
−
 
 
https://youtu.be/bSPWqeNO11M?si=HIzlxPjMfvGM5lxR
 
https://youtu.be/bSPWqeNO11M?si=HIzlxPjMfvGM5lxR
  
Line 69: Line 56:
  
 
==Video Solution by Dr. David==
 
==Video Solution by Dr. David==
−
 
 
https://youtu.be/RzPadkHd3Yc
 
https://youtu.be/RzPadkHd3Yc
  

Latest revision as of 21:09, 7 January 2025

Problem

What is the ones 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 by Central Valley Math Circle (Goes through full thought process)

https://youtu.be/-XcShDyuZIo

~mr_mathman

Video Solution 1 (Detailed Explanation)

https://youtu.be/jqsbMWhTYRg

~ ChillGuyDoesMath :)

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

~hsnacademy

Video Solution 4 (Quick and Easy!)

https://youtu.be/Ol1seWX0xHY

~Education, the Study of Everything

Video Solution by Interstigation

https://youtu.be/ktzijuZtDas&t=36

Video Solution by Daily Dose of Math

https://youtu.be/bSPWqeNO11M?si=HIzlxPjMfvGM5lxR

~Thesmartgreekmathdude

Video Solution by Dr. David

https://youtu.be/RzPadkHd3Yc

Video Solution by 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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2024_AMC_8_Problems/Problem_1&oldid=239341"