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

Revision as of 14:20, 27 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

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

Solution 2(tedious)

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

Solution 3

We only care about the unit's digits.

Thus, $2-2$ ends in $0$ , $0-2$ ends in $8$ , $8-2$ ends in $6$ , $6-2$ ends in $4$ , and $4-2$ ends in $\boxed{\textbf{(B) } 2}$ .

~MrThinker

Solution 4

Let $S$ 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:

$S\equiv 2 - 2 - 2 - 2- 2- 2 \equiv -8 \equiv -8 + 10\equiv \boxed{\textbf{(B) } 2} \pmod{10}.$

-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): $12-2-(2+2+2+2)=10-8=2$ Thus, we get the answer $\boxed{(B)}$

- U-King

Video Solution 1 (easy to digest) by Power Solve

https://www.youtube.com/watch?v=HE7JjZQ6xCk

Video Solution by NiuniuMaths (Easy to understand!)

https://www.youtube.com/watch?v=Ylw-kJkSpq8

~NiuniuMaths

Video Solution 2 by SpreadTheMathLove

https://www.youtube.com/watch?v=L83DxusGkSY

Video Solution by CosineMethod [🔥Fast and Easy🔥]

https://www.youtube.com/watch?v=IMqOfC9lZSo

@@ Line 13: / Line 13: @@
 ~ Dreamer1297
-==Solution 2==
+==Solution 2(tedious)==
 <cmath>222,222-22,222-2,222-222-22-2</cmath>

2024 AMC 8 (Problems • Answer Key • Resources)
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