Given an integer number $n$ (in base 10) and a base $b$, determine whether the representation of $n$ in base $b$, $(n)_b$ has a coefficient with value of $b-1$.

- n=21914, b=6
(21914)6 = 245242 => we have b-1=5 in the representation of (n)b
- n=50673, b=9
(50673)9 = 76453 => we don't have b-1=8 in the representation of (n)b

Is it possible to find a necessary and sufficient condition for existence of $b-1$ in $(n)_b$? Is there a fast algorithmic way to find the answer?

My try

If $b-1$ is in the coefficients of $(n)_b$ then there should be two integers $p$ and $r$ such that:

$(b-1)b^p + r = n$

we can convert this to Diophantine equation for $r$ and $q=b^p$:

$(b-1) q + r = n$

we can solve Diophantine equation and see if $q=b^p$ has an integer solution for $p$. But I am not sure if this is a sufficient condition for the original problem!?


The fastest way is likely going to be to compute the base-$b$ representation of $n$, and then scan over it to see whether it contains the desired digit. That is a linear-time algorithm.

Solving a Diophantine equation is far harder.


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