2025 AMC 8
Complete problem set with solutions and individual problem pages
Each of the even numbers is divided by . The remainders are recorded. Which histogram displays the number of times each remainder occurs?
- A.
- B.
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- D.
- E.
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(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 .
Solution 2
Writing down all the remainders gives us
In this list, there are numbers with remainder , numbers with remainder , numbers with remainder , numbers with remainder , numbers with remainder , numbers with remainder , and numbers with remainder . Manually computation of every single term can be avoided by recognizing the pattern alternates from to and there are terms. The only histogram that matches this is .
