2025 AMC 8 Problems/Problem 13

Problem

Each of the even numbers $2, 4, 6, \ldots, 50$ is divided by $7$ . The remainders are recorded. Which histogram displays the number of times each remainder occurs?

Solution

Writing down all of the numbers modulo $7$ , we have $2, 4, 6, 1, 3, 5, 0, 2, \ldots, 4, 6, 1$ . Notice how the the cycle $2, 4, 6, 1, 3, 5, 0$ repeats itself 3 times (because $\lfloor{\frac{50}{14}}\rfloor=3$ ). Then, we have $44$ , $46$ , $48$ , and $50$ remaining, which are $2$ , $4$ , $6$ , and $1$ mod 7, respectively. After adding them to our total count, the remainder $0$ occurs $3$ times, $1$ occurs $4$ times, $2$ occurs $4$ times, $3$ occurs $3$ times, $4$ occurs $4$ times, $5$ occurs $3$ times, and $6$ occurs $4$ times, which corresponds to histogram $\boxed{\text{(A)}}$ .

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Solution 2

Writing down all the remainders gives us

$2, 4, 6, 1, 3, 5, 0, 2, 4, 6, 1, 3, 5, 0, 2, 4, 6, 1, 3, 5, 0, 2, 4, 6, 1.$

In this list, there are $3$ numbers with remainder $0$ , $4$ numbers with remainder $1$ , $4$ numbers with remainder $2$ , $3$ numbers with remainder $3$ , $4$ numbers with remainder $4$ , $3$ numbers with remainder $5$ , and $4$ numbers with remainder $6$ . Manually computation of every single term can be avoided by recognizing the pattern alternates from $0, 2, 4, 6$ to $1, 3, 5$ and there are $25$ terms. The only histogram that matches this is $\boxed{\textbf{(A)}}$ .

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2025 AMC 8 Problems/Problem 13

Problem

Solution

Solution 2