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Difference between revisions of "2024 AMC 8 Problems/Problem 4"
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==Problem==
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==Problem 4==
 
When Yunji added all the integers from <math>1</math> to <math>9</math>, she mistakenly left out a number. Her incorrect sum turned out to be a square number. What number did Yunji leave out?
 
When Yunji added all the integers from <math>1</math> to <math>9</math>, she mistakenly left out a number. Her incorrect sum turned out to be a square number. What number did Yunji leave out?
  
 
<math>\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9</math>
 
<math>\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9</math>
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==Solution 1==
 
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The sum of the numbers from <math>1</math> to <math>9</math> is <cmath>1 + 2 + 3 + \cdots + 9 = \frac{9(10)}{2} = 45.</cmath> Denote the number left out when adding to be <math>x</math>. Thus, <math>45 - x</math> is a perfect square. We also know that <math>x</math> must be between <math>1</math> and <math>9</math> inclusive. We can now start to find the closest perfect square to 45. We can immediately see that 36 is the closest, so we can just subtract both numbers to get our final answer of <math>\boxed{\textbf{(E) }9}</math>.
 
  
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~ Nivaar
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==Solution==
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Adding numbers 1 through 9 gives us 45. This was her expected sum, but what she got was a perfect square. Since she got that perfect square sum by forgetting a number, that sum is less than 45. The square number right under 45 is 36. So 45 - 36 = 9. So the solution is E[9].
  
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==Solution 2==
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~That_One_Dude
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Since we have all the answer choices, we can check and see which one works. Testing, we have that leaving out <math>9</math> works, so the answer is <math>\boxed{\textbf{(E) }9}</math>.
 
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~andliu766
 
  
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-rnatog337
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== Video Solution 1 (Detailed Explanation) 🚀⚡📊 ==
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==Solution 3==
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Youtube Link ⬇️
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Recall from AMC12A 2022 Problem 16, that <math>1+2+\dots+8 = 6^2</math>. Hence removing <math>9</math> works and our answer is <math>\boxed{\textbf{(E) }9}</math>.
 
  
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-SahanWijetunga
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https://youtu.be/5mCSOiqX8sA
  
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==Video Solution 1 (easy to digest) by Power Solve==
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~ ChillGuyDoesMath :)
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==Video Solution by Central Valley Math Circle (Goes through the full thought process)==
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https://youtu.be/dMX5_MXVkYs
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 +
~mr_mathman
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== Video by MathTalks_Now ==
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https://www.youtube.com/watch?v=crn37TRMLv4
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 +
-rc1219
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==Video Solution by Math-X (First fully understand the problem!!!)==
 +
https://youtu.be/BaE00H2SHQM?si=tzbzeCdEfXqROciK&t=907
 +
 
 +
~Math-X
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 +
==Video Solution (A Clever Explanation You’ll Get Instantly)==
 +
https://youtu.be/5ZIFnqymdDQ?si=bxVKQiul4AjzVUgC&t=331
 +
 
 +
~hsnacademy
 +
 
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==Video Solution 1 (Simple and Fast) by Parshwa==
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https://youtu.be/Qrf4GTDjjXs
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==Video Solution 2 (easy to digest) by Power Solve==
 
https://youtu.be/HE7JjZQ6xCk?si=sTC7YNSmfEOMe4Sn&t=179
 
https://youtu.be/HE7JjZQ6xCk?si=sTC7YNSmfEOMe4Sn&t=179
  
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== Video Solution by CosineMethod [🔥Fast and Easy🔥]==
 
== Video Solution by CosineMethod [🔥Fast and Easy🔥]==
  
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https://www.youtube.com/watch?v=9v5q5DxeriM
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==Video Solution by Intersigation==
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https://youtu.be/ktzijuZtDas&t=232
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==Video Solution by Daily Dose of Math (Understandable, Logical, and Speedy)==
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https://youtube/bSP
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qeNO11M?si=HIzlxPjMfvGM5lxR
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~Thesmartgreekmathdude
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==Video Solution by WhyMath==
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https://youtu.be/PCvkqxz0f0I
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2024|num-b=3|num-a=5}}
 
{{AMC8 box|year=2024|num-b=3|num-a=5}}
 
{{MAA Notice}}
 
{{MAA Notice}}
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[[Category:Introductory Number Theory Problems]]

Latest revision as of 10:11, 31 July 2025

Problem 4

When Yunji added all the integers from $1$ to $9$, she mistakenly left out a number. Her incorrect sum turned out to be a square number. What number did Yunji leave out?

$\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9$

Solution

Adding numbers 1 through 9 gives us 45. This was her expected sum, but what she got was a perfect square. Since she got that perfect square sum by forgetting a number, that sum is less than 45. The square number right under 45 is 36. So 45 - 36 = 9. So the solution is E[9].

~That_One_Dude

Video Solution 1 (Detailed Explanation) 🚀⚡📊

Youtube Link ⬇️

https://youtu.be/5mCSOiqX8sA

~ ChillGuyDoesMath :)


Video Solution by Central Valley Math Circle (Goes through the full thought process)

https://youtu.be/dMX5_MXVkYs

~mr_mathman

Video by MathTalks_Now

https://www.youtube.com/watch?v=crn37TRMLv4

-rc1219

Video Solution by Math-X (First fully understand the problem!!!)

https://youtu.be/BaE00H2SHQM?si=tzbzeCdEfXqROciK&t=907

~Math-X

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

https://youtu.be/5ZIFnqymdDQ?si=bxVKQiul4AjzVUgC&t=331

~hsnacademy

Video Solution 1 (Simple and Fast) by Parshwa

https://youtu.be/Qrf4GTDjjXs

Video Solution 2 (easy to digest) by Power Solve

https://youtu.be/HE7JjZQ6xCk?si=sTC7YNSmfEOMe4Sn&t=179

Video Solution by Math-X (First fully understand the problem!!!)

https://youtu.be/BaE00H2SHQM?si=9ZUxEGmGam7il9xr&t=907

~Math-X


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🔥]

Video Solution by Intersigation

https://youtu.be/ktzijuZtDas&t=232

Video Solution by Daily Dose of Math (Understandable, Logical, and Speedy)

https://youtube/bSP qeNO11M?si=HIzlxPjMfvGM5lxR

~Thesmartgreekmathdude

Video Solution by WhyMath

https://youtu.be/PCvkqxz0f0I

See Also

2024 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

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