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