AMC8 2025

AMC8 2025 · Q13

AMC8 2025 · Q13. It mainly tests Remainders & modular arithmetic, Sequences in number theory (remainders patterns).

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?

每个偶数$2, 4, 6, \ldots, 50$除以$7$。记录余数。哪个直方图显示了每个余数出现的次数？

(A)

(B)

(C)

(D)

(E)

Answer

Correct choice: (A)

正确答案：（A）

Solution

Let's take the numbers 2 through 14 (evens). The remainders will be 2, 4, 6, 1, 3, 5, and 0. This sequence keeps repeating itself over and over. We can take floor(50/14) = 3, so after the number 42, every remainder has been achieved 3 times. However, since 44, 46, 48, and 50 are left, the remainders of those will be 2, 4, 6, and 1 respectively. The only histogram in which those 4 numbers are set at 4 is histogram $\boxed{\textbf{(A)}}$.

取2到14（偶数）的数字。余数依次为2、4、6、1、3、5、0。这个序列不断重复。我们有$\lfloor50/14\rfloor = 3$，所以到42后，每个余数出现3次。然而，还剩44、46、48、50，它们的余数分别是2、4、6、1。只有那个4个余数为4的直方图是$\boxed{\textbf{(A)}}$。

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