Difference between revisions of "2025 AMC 8 Problems/Problem 13"
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==Video Solution by Thinking Feet== | ==Video Solution by Thinking Feet== | ||
https://youtu.be/PKMpTS6b988 | https://youtu.be/PKMpTS6b988 | ||
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| + | ==See Also== | ||
| + | {{AMC8 box|year=2025|num-b=12|num-a=14}} | ||
| + | {{MAA Notice}} | ||
Revision as of 20:06, 30 January 2025
Problem
Each of the even numbers 
is divided by 
. The remainders are recorded. Which histogram displays the number of times each remainder occurs?
Solution
Writing down all of the numbers modulo , we have 
. Notice how the the cycle 
repeats itself 3 times (because 
). Then, we have 
, 
, 
, and 
remaining, which are 
, 
, 
, and 
mod 7, respectively. After adding them to our total count, the remainder 
occurs 
times, occurs times, occurs times, occurs times, occurs times, 
occurs times, and occurs times, which corresponds to histogram 
.
~mrtnvlknv
Solution 2
Writing down all the remainders gives us
![\[2, 4, 6, 1, 3, 5, 0, 2, 4, 6, 1, 3, 5, 0, 2, 4, 6, 1, 3, 5, 0, 2, 4, 6, 1.\]](http://latex.artofproblemsolving.com/e/a/d/ead2ad7bd4fcb3c595a09558d722006e01c9b5ee.png)
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 
.
~alwaysgonnagiveyouup
Video Solution by Thinking Feet
See Also
| 2025 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
