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

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==Problem==
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==Problem 1==
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>
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What is the ones digit of: <cmath>222{,}222-22{,}222-2{,}222-222-22-2?</cmath>
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<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 <cmath>222,222-(22,222+2,222+222+22+2).</cmath>
 
+
 
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>.
 
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.
 
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.
  
i am smart
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==Solution 2==
  
~ Dreamer1297
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<math>222,222-22,222 = 200,000</math>
 
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<math>200,000 - 2,222 = 197778</math>
==Solution 2(Tedious)==
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<math>197778 - 222 = 197556</math>
 
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<math>197556 - 22 = 197534</math>
<cmath>222,222-22,222-2,222-222-22-2</cmath>
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<math>197534 - 2 = 1957532
<cmath> = 200,000 - 2,222 - 222 - 22 - 2 </cmath>
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</math>
<cmath> = 197778 - 222 - 22 - 2 </cmath>
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So our answer is <math>\boxed{\textbf{(B) } 2}</math>.
<cmath> = 197556 - 22 - 2 </cmath>
 
<cmath> = 197534 - 2 </cmath>
 
<cmath> = 197532 </cmath>
 
This means the ones digit is <math>\boxed{\textbf{(B)} \hspace{1 mm} 2}</math>
 
 
 
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.
 
<math>\newline</math>
 
~ nikhil
 
~ CXP
 
~ Nivaar
 
  
 
==Solution 3==
 
==Solution 3==
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We only care about the unit's digits.
 
We only care about the unit's digits.
  
Thus, <math>2-2</math> ends in <math>0</math>, <math>0-2</math> 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{(A) } -098765432345q67w565374865368769chvdfhb}</math>.
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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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-unknown
  
~iasdjfpawregh
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minor edits by Fireball9746
  
 
==Solution 4==
 
==Solution 4==
  
Let <math>S</math> be equal to the expression at hand. We reduce each term modulo <math>10</math> to find the units digit of each term in the expression, and thus the units digit of the entire thing:  
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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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<cmath>(12-2)-(2+2+2+2)=10-8=2</cmath>
<cmath>S\equiv 2 - 2 - 2 - 2- 2- 2 \equiv -8 \equiv -8 + 10\equiv \boxed{\textbf{(B) } 2} \pmod{10}.</cmath>
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Thus, we get the answer <math>\boxed{(B)}</math>
  
-Benedict T (countmath1)
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==Video Solution (MATH-X)==
 +
https://youtu.be/BaE00H2SHQM?si=O0O0g7qq9AbhQN9I&t=130
  
 +
~Math-X
  
 +
==Video Solution (A Clever Explanation You’ll Get Instantly)==
 +
https://youtu.be/5ZIFnqymdDQ?si=IbHepN2ytt7N23pl&t=53
  
==Solution 5==
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~hsnacademy
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 <math>\boxed{(B)}</math>
 
 
 
- U-King
 
 
 
==Solution 6(fast)==
 
Let's only look at the ones digit: 2 - (2+2+2+2+2) = 2 - 0 = 2
 
Therefore, ones digit is 2  <math>\boxed{(B)}</math>
 
  
- thebanker88
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==Video Solution  (Quick and Easy!)==
 +
https://youtu.be/Ol1seWX0xHY
  
==Video Solution by Math-X (First fully understand the problem!!!)==
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~Education, the Study of Everything
https://youtu.be/BaE00H2SHQM?si=ibOBlVvzgRSNsrOA&t=130
 
  
~Math-X
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==Video Solution by Interstigation==
 +
https://youtu.be/ktzijuZtDas&t=36
  
==Video Solution 1 (easy to digest) by Power Solve==
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==Video Solution by Daily Dose of Math==
https://www.youtube.com/watch?v=HE7JjZQ6xCk
 
  
==Video Solution by NiuniuMaths (Easy to understand!)==
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https://youtu.be/bSPWqeNO11M?si=HIzlxPjMfvGM5lxR
https://www.youtube.com/watch?v=Ylw-kJkSpq8
 
  
~NiuniuMaths
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~Thesmartgreekmathdude
  
==Video Solution 2 by SpreadTheMathLove==
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==Video Solution by Dr. David==
https://www.youtube.com/watch?v=L83DxusGkSY
 
  
== Video Solution by CosineMethod [🔥Fast and Easy🔥]==
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https://youtu.be/RzPadkHd3Yc
  
https://www.youtube.com/watch?v=IMqOfC9lZSo
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==Video Solution by WhyMath==
 +
https://youtu.be/i4mcj3jRTxM
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2024|before=First Problem|num-a=2}}
 
{{AMC8 box|year=2024|before=First Problem|num-a=2}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 21:35, 16 December 2024

Problem 1

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.

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 unit's 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 $\boxed{\textbf{(B) } 2}$.

-unknown

minor edits by Fireball9746

Solution 4

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 $\boxed{(B)}$

Video Solution (MATH-X)

https://youtu.be/BaE00H2SHQM?si=O0O0g7qq9AbhQN9I&t=130

~Math-X

Video Solution (A Clever Explanation You’ll Get Instantly)

https://youtu.be/5ZIFnqymdDQ?si=IbHepN2ytt7N23pl&t=53

~hsnacademy

Video Solution (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 (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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