Problem:
Let N be the number of ways to write 2010 in the form
2010=a3⋅103+a2⋅102+a1⋅10+a0
where the ai 's are integers, and 0≤ai≤99. An example of such a representation is 1⋅103+3⋅102+67⋅101+40⋅100. Find N.
Solution:
Write ai=10bi+ci, where bi,ci∈{0,1,2,…,7,8,9}; if bi and ci are chosen in this way, they determine a unique acceptable ai.
Let m=b3⋅103+b2⋅102+b1⋅101+b0⋅100, and n=c3⋅103+c2⋅102+ c1⋅101+c0⋅100, and write the representation as
2010=(10b3+c3)103+(10b2+c2)102+(10b1+c1)101+(10b0+c0)100=10m+n
The number of such representations is the number of ways to write 2010 as 10m+n, where m and n are nonnegative integers. That is, m∈ {0,1,…,201} and n=2010−10m. Thus N=202.
The problems on this page are the property of the MAA's American Mathematics Competitions