2024 AMC 8 Problems/Problem 4

Problem

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 1

The sum of the numbers from $1$ to $9$ is $1 + 2 + 3 + \cdots + 9 = \frac{9(10)}{2} = 45.$ Denote the number left out when adding to be $x$ . Thus, $45 - x$ is a perfect square. We also know that $x$ must be between $1$ and $9$ inclusive, so the answer is $\boxed{\textbf{(E) }9}$ .

Solution 2

Since we have all the answer choices, we can check and see which one works. Testing, we have that leaving out $9$ works, so the answer is $\boxed{\textbf{(E) }9}$ . ~andliu766

-rnatog337

Solution 3

Recall from AMC12A 2022 Problem 16, that $1+2+\dots+8 = 6^2$ . Hence removing $9$ works and our answer is $\boxed{\textbf{(E) }9}$ .

-SahanWijetunga

Video Solution 1 (easy to digest) by Power Solve

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

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=9v5q5DxeriM

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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