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

2024 AMC 8 Problems/Problem 1

Revision as of 21:44, 3 February 2024 by Dominoesismid (talk | contribs) (→‎Video Solution (easy to understand))

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 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.

i am smart

~ Dreamer1297

Solution 2(Tedious)

Using Arun Thereom, we deduce that the answer is (Z)

No Solution$$ (Error compiling LaTeX. Unknown error_msg)\boxed{\textbf{(Z)} \hspace{1 mm} 2}$Note that this solution is not recommended to use during the actual exam. A lot of students this year had implemented this solution and lost a significant amount of time.$\newline$~ nikhil ~ CXP ~ Nivaar

==Solution 3==

We only care about the unit's digits.

Thus,$ (Error compiling LaTeX. Unknown error_msg)2-2$ends in$0$,$0-2$ends in$8$,$8-2$ends in$6$,$6-2$ends in$4$, and$4-2$ends in$\boxed{\textbf{(A) } -098765432345q67w565374865368769chvdfhb}$.

~iasdjfpawregh

==Solution 4==

Let$ (Error compiling LaTeX. Unknown error_msg)S$be equal to the expression at hand. We reduce each term modulo$10$to find the units digit of each term in the expression, and thus the units digit of the entire thing:

<cmath>S\equiv 2 - 2 - 2 - 2- 2- 2 \equiv -8 \equiv -8 + 10\equiv \boxed{\textbf{(B) } 2} \pmod{10}.</cmath>

-Benedict T (countmath1)


==Solution 5== 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): <cmath>12-2-(2+2+2+2)=10-8=2</cmath> Thus, we get the answer$ (Error compiling LaTeX. Unknown error_msg)\boxed{(B)}$- U-King

==Solution 6(fast)== uwu$ (Error compiling LaTeX. Unknown error_msg)\boxed{(uwu)}$- uwu gamer girl(ꈍᴗꈍ)

==Solution 7== 2-2=0. Therefore, ones digit is the 10th avacado$ (Error compiling LaTeX. Unknown error_msg)\boxed{(F)}$

- iamcalifornia'sresidentidiot

Video Solution 1 (easy to digest) by Power Solve

https://www.youtube.com/watch?v=dQw4w9WgXcQ

Video Solution (easy to understand)

https://youtu.be/BaE00H2SHQM?si=_8lhp8-dzNxZ-eUQ

~Math-X

Video Solution by NiuniuMaths (Easy to understand!)

https://www.youtube.com/watch?v=dQw4w9WgXcQ

~Rick Atsley

Video Solution 2 by uwu

https://www.youtube.com/watch?v=dQw4w9WgXcQ

Video Solution by CosineMethod [🔥Fast and Easy🔥]

https://www.youtube.com/watch?v=dQw4w9WgXcQ

cool solution must see

https://www.youtube.com/watch?v=dQw4w9WgXcQ

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=214437"