Problem:
Define Ln​ as the least common multiple of all the integers from 1 to n inclusive. There is a unique integer h such that
11​+21​+31​+⋯+171​=L17​h​
What is the remainder when h is divided by 17?
Answer Choices:
A. 1
B. 3
C. 5
D. 7
E. 9
Solution:
Note that
1+21​+⋯+161​=L16​m​
for some integer m, so the given sum is
L16​m​+171​=L17​h​
Now L17​=17L16​, so it follows that h=17m+L16​. Therefore h≡L16​(mod17). Note that L16​=24⋅32⋅5⋅7⋅11⋅13. These factors modulo 17 satisfy 24≡−1,5⋅7≡1,9⋅11≡−3, and 13≡−4, so the product is congruent to (−1)(1)(−3)(−4)=−12≡(C)5​(mod17).
The problems on this page are the property of the MAA's American Mathematics Competitions