Problem:
For each integer n≥4, let an​ denote the base- n number 0.133n​. The product a4​a5​…a99​ can be expressed as n!m​, where m and n are positive integers and n is as small as possible. What is the value of m ?
Answer Choices:
A. 98
B. 101
C. 132
D. 798
E. 962
Solution:
Note that n3an​=133⋅133n​=an​+n2+3n+3, so
an​=n31n2+3n+3​=n(n3−1)(n+1)3−1​
Therefore
a4​a5​a99​=4(43−1)53−1​5(53−1)63−1​99(993−1)10031​=99!3!​43−11003−1​=99!6​6399(1002+100+1)​=(21)(98!)(2)(10,101)​=98!962​​​
The problems on this page are the property of the MAA's American Mathematics Competitions