Thursday, November 10th

Problem
A number is said decreasing if its digits are non-increasing from left to right. Are there integers n such that 16^n is decreasing? 


Solution 
Since the sum of the digits of $16^n$ is $6n+1$ and $16^n$ has $\lfloor n\log(16)\rfloor$ digits, in order to have a decreasing power of 16, its digit sum has to be at least $6\lfloor n\log(16)\rfloor \ge 6.2n$, hence it is not possible to have such number.