Problem:
Find the three-digit positive integer abc whose representation in base nine is bcanine , where a,b, and c are (not necessarily distinct) digits.
Solution:
Writing the problem conditions out algebraically gives us
100a+10b+c=81b+9c+a99a=71b+8c
We can take mod 71, which gives us
28a≡8c(mod8)7a≡2c(mod8)
We can easily see that (a,c)=(2,7) and b=2.
Thus, abc=227.
The problems on this page are the property of the MAA's American Mathematics Competitions