Problem:
For which of the following integers b is the base-b number 2021b​−221b​ not divisible by 3?
Answer Choices:
A. 3
B. 4
C. 6
D. 7
E. 8
Solution:
Note that
2021b​−221b​=2b3+2b+1−(2b2+2b+1)=2b3−2b2=2b2(b−1).
This number is divisible by 3 if and only if either b or b−1 is divisible by 3 . Because neither 8 nor 8−1 is divisible by 3, the (E)​ base-eight number 2021eight ​−221eight ​ is not divisible by 3. For all the other given choices for b, either b or b−1 is divisible by 3.
The problems on this page are the property of the MAA's American Mathematics Competitions