Problem:
Let n be a 5-digit number, and let q and r be the quotient and remainder, respectively, when n is divided by 100. For how many values of n is q+r divisible by 11?
Answer Choices:
A. 8180
B. 8181
C. 8182
D. 9000
E. 9090
Solution:
Note that n=100q+r=q+r+99q. Hence q+r is divisible by 11 if and only if n is divisible by 11. Since 10,000≤n≤99,999, there are
⌊1199999​⌋−⌊119999​⌋=9090−909=8181
such numbers.
Answer: B​.
The problems on this page are the property of the MAA's American Mathematics Competitions