AMC8 2024

AMC8 2024 · Q4

AMC8 2024 · Q4. It mainly tests Averages (mean), Perfect squares & cubes.

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?

当 Yunji 将从 $1$ 到 $9$ 的所有整数相加时，她错误地遗漏了一个数。她的错误和恰好是一个完全平方数。Yunji 遗漏了哪个数？

(A) 5 5

(B) 6 6

(D) 8 8

(E) 9 9

Answer

Correct choice: (E)

正确答案：（E）

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 $\boxed{\textbf{(E)}\ 9}$.

1 到 9 的和是 45。这是她预期的和，但她得到的是一个完全平方数。由于她忘记了一个数，这个和小于 45。45 下方的平方数是 36。所以 45 - 36 = 9。因此答案是 $\boxed{\textbf{(E)}\ 9}$。

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