Problem:
For all positive integers n less than 2002 , let
an=⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧11, if n is divisible by 13 and 1413, if n is divisible by 14 and 1114, if n is divisible by 11 and 130, otherwise
Calculate ∑n=12001an.
Answer Choices:
A. 448
B. 486
C. 1560
D. 2001
E. 2002
Solution:
Since 2002=11⋅13⋅14, we have
an=⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧11, if n=13⋅14⋅i, where i=1,2,…,1013, if n=14⋅11⋅j, where j=1,2,…,1214, if n=11⋅13⋅k, where k=1,2,…,130, otherwise
Hence ∑n=12001an=11⋅10+13⋅12+14⋅13=448.
The problems on this page are the property of the MAA's American Mathematics Competitions