$11 \cdot 30 \quad N$ is an integer, its $b$-ary representation is 777. Find the smallest positive integer $b$, such that $N$ is the fourth power of an integer in decimal.
(9th Canadian Mathematics Competition, 1977)
[Solution] This problem is equivalent to finding the smallest positive integer $b$ such that the equation
$$
7 b^{2}+7 b+7=x^{4}
$$

has an integer solution for $x$.
Since 7 is a prime number, from equation (1), 7 is a divisor of $x$, so let $x=7 k$, then equation (1) becomes
$$
b^{2}+b+1=7^{3} k^{4} .
$$

The smallest $b$ occurs when $k$ is the smallest.
Taking $k=1$, at this point we have
$$
\begin{array}{l}
b^{2}+b+1=343, \\
b^{2}+b-342=0 .
\end{array}
$$

At this point we have
$$
(b-18)(b+19)=0 \text {. }
$$

Solving for the positive integer
$$
b=18 \text {. }
$$

We use $(A)_{k}$ to denote a number in base $k$, then we have
$$
(777)_{18}=\left(7^{4}\right)_{10} \text {. }
$$