1977 Canadian MO Problems/Problem 3


$N$ is an integer whose representation in base $b$ is $777.$ Find the smallest positive integer $b$ for which $N$ is the fourth power of an integer.


Rewriting $N$ in base $10,$ $N=7(b^2+b+1)=a^4$ for some integer $a.$ Because $7\mid a^4$ and $7$ is prime, $a^4 \ge 7^4.$ Since we want to minimize $b,$ we check to see if $a=7$ works.

When $a=7,$ $b^2+b+1=7^3.$ Solving this quadratic, $b = 18$.

1977 Canadian MO (Problems)
Preceded by
Problem 2
1 2 3 4 5 6 7 8 Followed by
Problem 4