Difference between revisions of "1977 Canadian MO Problems/Problem 3"

Revision as of 13:55, 5 July 2008

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, $a \ge 7^4.$ Since we want to minimize $b,$ we check to see if $a=7^4$ works.

When $a=7^4,$ $b^2+b+1=7^3.$ 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

Revision as of 21:49, 17 November 2007 (view source) Temperal (talk \| contribs) (problem) ← Older edit		Revision as of 13:55, 5 July 2008 (view source) Archimedes1 (talk \| contribs) (Moved to Intermediate category) Newer edit →
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	{{Old CanadaMO box\|num-b=2\|num-a=4\|year=1977}}		{{Old CanadaMO box\|num-b=2\|num-a=4\|year=1977}}

−	[[Category:~~Olympiad~~ Number Theory Problems]]	+	[[Category:Intermediate Number Theory Problems]]

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Difference between revisions of "1977 Canadian MO Problems/Problem 3"

Revision as of 13:55, 5 July 2008

Problem

Solution