2024 AMC 8 Problem 4

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To help you master 2024 AMC 8 Problem 4, this page includes:

• The original AMC problem to try, plus the correct answer

• Clear, step-by-step solutions using different methods

• Key ideas, useful formulas, and takeaways you can apply to similar problems

• Related problems from past AMC exams, plus new practice problems that build the same skills

Want more practice? Scroll down to download Think Academy’s AMC 8 practice packs and keep building your AMC 8 skills.

2024 AMC8 Problem 4

Solution

Solution 1

Direct Bounding Method

Calculate the intended sum of all the integers from 1 to 9. By subtracting a variable for the missing number, you can find a narrow range of possible sums, making it easy to identify the only perfect square.

Step 1: Calculate the intended total sum

First, we find the total sum if no numbers were left out. We can do this by using the formula for the sum of the first positive integers, or simply by adding them up directly.

$$S=\frac{9(9+1)}{2}$$

$$S=\frac{90}{2}=45$$

Step 2: Express the incorrect sum algebraically

Let the missing number be a variable. Since Yunji left out exactly one number from the list, this variable must be an integer from 1 to 9. Her incorrect sum is simply the true sum minus this missing number.

$$\text{Incorrect sum}=45-x$$

$$1\le x\le 9$$

Step 3: Find the possible range for the incorrect sum

We find the range of possible values for the incorrect sum by subtracting the largest and smallest possible missing numbers from the total sum.

$$45-9\le 45-x\le 45-1$$

$$36\le \text{Incorrect sum}\le 44$$

Step 4: Identify the perfect square

The problem states that the incorrect sum is a perfect square. We evaluate the squares of nearby integers to find the only perfect square that falls strictly inside our bounded range.

$$5^2=25$$

$$6^2=36$$

$$7^2=49$$

$$36\le 36\le 44$$

Step 5: Determine the missing number

Since 36 is the only valid perfect square in the range, we set the expression for the incorrect sum equal to 36 and solve for the missing number.

$$45-x=36$$

$$x=9$$

Correct answer: E. $9$
$\boxed{9}$ (Choice E) Final Check: The true sum is 45. If Yunji leaves out 9, her incorrect sum is 45 – 9 = 36. Since 36 is 6 squared, it is a perfect square, completely satisfying the problem’s conditions.

Solution 2

Multiple-Choice Shortcut

Since this is a multiple-choice question, the missing number must be one of the five provided options. You can test each option by subtracting it from the true total sum to see which resulting difference is a perfect square.

Step 1: Find the correct total sum

Just as in the direct method, we first determine what the sum of the numbers from 1 to 9 should have been.

$$1+2+3+4+5+6+7+8+9=45$$

Step 2: Test the answer choices

Subtract each given answer choice from the correct total sum to find what the incorrect sum would be in each scenario. We are looking for a difference that is a perfect square.

$$\text{A:}45-5=40$$

$$\text{B:}45-6=39$$

$$\text{C:}45-7=38$$

$$\text{D:}45-8=37$$

$$\text{E:}45-9=36$$

Step 3: Identify the correct choice

We check the results to see which one is a perfect square. The only perfect square among these results is 36, which is generated by Choice E.

$$36=6^2$$

Correct answer: E. $9$
$\boxed{9}$ (Choice E) Final Check: A quick visual check confirms that none of the other resulting sums (40, 39, 38, 37) are perfect squares, so 9 is definitively the only correct answer.

Similar Problems in Real Exams

The following five AMC 8 problems are closely related to AMC 8 2024 Question 4. These real AMC 8 problems test the same key concept—Number Theory. Click any problem to learn and practice, then continue to the adapted practice section later in this article for more problems to try.

2018 AMC 8 Problem 25

2016 AMC 8 Problem 7

2010 AMC 8 Problem 24

2009 AMC 8 Problem 17

2026 AMC 8 Problem 8

Adapted Problems to Practice

What Formulas Do You Need for This Problem?

• Sum of the First n Positive Integers:

n is the largest integer in the sequence starting from 1. It allows for a quick and error-free calculation of the sum of the numbers from 1 to 9 without having to manually add all nine numbers together.

• Perfect Square:

An integer N is a perfect square if N = k^2 for some integer k. The problem states the incorrect sum is a square number, which serves as the core condition restricting the possible values of the sum.

Key Takeaway: Remember This Strategy for Next Time

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