AMC 8 2025 Problem 13
AMC8 2025 年第 13 题
余数与同余★★★☆☆
Each of the even numbers is divided by 7 . The remainders are recorded. Which histogram displays the number of times each remainder occurs?
将偶数2、4、6、……、50分别除以7，并记录下余数。请问下方哪个直方图显示了每个余数出现的次数？
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
A
解析 · Solution
Solution 1
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() = 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.
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.
First, we find all of the multiples of 7 that are even, and are therefore, in this list. Knowing that 7 is odd, and that odd+odd=even, we can find all of the even multiples of 7 by simply finding all of the multiples of 14 that fit on this list. Doing this, we end up with $14, 28, 42.$ Now, we can clearly see that there are only 3 numbers in this list with 0 as a remainder. This disproves 2 of our 5 answers immediately. Since our remaining answers are identical until we reach 3 as a remainder, we can skip right to there. Now, we need to find all even numbers that leave a remainder of 3 when divided by 7. To do this, we add 3 to all ODD multiples of 7 because odd+odd=even. This gives us $10, 24, 38.$
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