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"

(→‎Solution 2)
(→‎Solution 1)
Line 6: Line 6:
  
 
<cmath>222,222-22,222-2,222-222-22-2</cmath>
 
<cmath>222,222-22,222-2,222-222-22-2</cmath>
−
<cmath> \equiv 2-2-2-2-2-2</cmath>
+
<cmath> = 200,000 - 2,222 - 222 - 22 - 2 </cmath>
−
<cmath> \equiv -8</cmath>
+
<cmath> = 197778 - 222 - 22 - 2 </cmath>
−
<cmath> \equiv 2</cmath>  
+
<cmath> = 197556 - 22 - 2 </cmath>
−
So the solution is <math>(B) </math> <math> 2</math>
+
<cmath> = 197534 - 2 </cmath>
−
 
+
<cmath> = 197532 </cmath>
−
~nikhil
+
This means the ones digit is <math>\boxed{(B) \hspace{1 mm} 2}</math>
 +
<math>\newline</math>
 +
~ nikhil
 
~ CXP
 
~ CXP
  

Revision as of 15:01, 25 January 2024

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

\[222,222-22,222-2,222-222-22-2\] \[= 200,000 - 2,222 - 222 - 22 - 2\] \[= 197778 - 222 - 22 - 2\] \[= 197556 - 22 - 2\] \[= 197534 - 2\] \[= 197532\] This means the ones digit is $\boxed{(B) \hspace{1 mm} 2}$ $\newline$ ~ nikhil ~ CXP

Solution 2

We can rewrite the expression as \[222,222-(22,222+2,222+222+22+2)\].

We note that the units digit of the addition is $0$ because all the units digits of the five numbers are $2$ and $5*2=10$, which has a units digit of $0$.

Now, we have something with a units digit of $0$ subtracted from $222,222$. The units digit of this expression is obviously $2$, and we get $\boxed{B}$ as our answer.

~ Dreamer1297

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2024_AMC_8_Problems/Problem_1&oldid=212806"