1977 Canadian MO Problems/Problem 3

Problem

$N$ 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.

Solution

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

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1977 Canadian MO Problems/Problem 3

Problem

Solution