2025 AMC 8 Problem 6

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To help you master 2025 AMC 8 Problem 6, 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.

2025 AMC8 Problem 6

Solution

Solution 1

Total Sum Method

Find the sum of all five numbers. Then, test each possible erased number by subtracting it from the total and checking if the result is divisible by 4.

Step 1: Calculate the total sum

First, we find the sum of all five numbers on Sekou’s list. We can add them directly, or note that the average of these five consecutive numbers is 17.

$$15+16+17+18+19=85$$

Step 2: Set up the requirement

Let the erased number be represented by a variable. The problem states that the sum of the remaining four numbers must be a multiple of 4.

$$\text{Remaining Sum}=85-x$$

Step 3: Test each possible erased number

We subtract each of the given choices from the total sum to see which result can be divided evenly by 4. Notice that if the remaining sum is an odd number, it cannot be a multiple of 4.

$$85-15=70(70=4\times 17.5\to \text{No})$$

$$85-16=69(\text{Odd number}\to \text{No})$$

$$85-17=68(68=4\times 17\to \text{Yes})$$

$$85-18=67(\text{Odd number}\to \text{No})$$

$$85-19=66(66=4\times 16.5\to \text{No})$$

Step 4: Identify the correct number

Since erasing 17 is the only option that leaves a multiple of 4, it must be the number Sekou erased.

Correct answer: C. $17$
The erased number is $17$, which corresponds to choice C. We can verify this by checking the sum of the remaining numbers: $15+16+18+19=68$. Since $68÷4=17$, the condition is perfectly satisfied.

Solution 2

Remainder Method

Instead of calculating the full sum, we focus only on the remainders when the numbers are divided by 4. The erased number must have the same remainder as the total sum so that removing it leaves a remainder of 0.

The remainder of a sum is equal to the sum of the individual remainders. This allows us to work with very small numbers.

Step 1: Find the remainder of each number

We divide each number on the list by 4 and record only its remainder.

$$15÷4=3\text{R}3$$

$$16÷4=4\text{R}0$$

$$17÷4=4\text{R}1$$

$$18÷4=4\text{R}2$$

$$19÷4=4\text{R}3$$

Step 2: Find the remainder of the total sum

We add the individual remainders together. Then, we find the remainder of this new, smaller sum when divided by 4.

$$\text{Sum of remainders}=3+0+1+2+3=9$$

$$9÷4=2\text{R}1$$

Step 3: Determine the required remainder for the erased number

The total sum is 1 more than a multiple of 4. To make the remaining sum a perfect multiple of 4, the number we erase must also be exactly 1 more than a multiple of 4. Therefore, the erased number must have a remainder of 1.

Step 4: Identify the erased number

Looking back at our list of remainders from the first step, we check which number produced the required remainder.

$$17÷4=4\text{R}1$$

Correct answer: C. $17$
The erased number is $17$, which corresponds to choice C. Because both the total sum ($85$) and $17$ leave a remainder of $1$ when divided by $4$, their difference leaves a remainder of $0$.

Similar Problems in Real Exams

The following five AMC 8 problems are closely related to AMC 8 2025 Question 6. These real AMC 8 problems test the same key concept—Remainders & Modular Arithmetic. Click any problem to learn and practice, then continue to the adapted practice section later in this article for more problems to try.

2025 AMC 8 Problem 13

2019 AMC 8 Problem 14

2018 AMC 8 Problem 21

2016 AMC 8 Problem 5

2013 AMC 8 Problem 1

Adapted Problems to Practice

What Formulas Do You Need for This Problem?

• Properties of Remainders (Modular Arithmetic):

Applies whenever you need to check if a sum of multiple numbers is divisible by a specific integer. We can determine if the remaining four numbers add up to a multiple of 4 without actually adding them, by just operating on their remainders.

• Subtracting Remainders:

Applies when removing a part from a whole to achieve exact divisibility. The total sum had a remainder of 1 when divided by 4. Removing the number 17 (which also has a remainder of 1) perfectly eliminates the extra amount.

Key Takeaway: Remember This Strategy for Next Time

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