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 could rigorously solve the values of $x$ such that $45 - x$ is a perfect square. However, since this is the AMC 8, we can simply try each option. When $x = 9$ , we see that $45 - 9 = 36$ , a perfect square. Therefore, our answer is $\boxed{\textbf{(E) }9}$ .

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2024 AMC 8 Problems/Problem 4

Problem

Solution 1