Difference between revisions of "1977 Canadian MO Problems/Problem 3"
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== Problem == | == Problem == | ||
− | <math> | + | <math>N</math> is an integer whose representation in base <math>b</math> is <math>777.</math> Find the smallest positive integer <math>b</math> for which <math>N</math> is the fourth power of an integer. |
== Solution == | == Solution == | ||
− | Rewriting <math> | + | Rewriting <math>N</math> in base <math>10,</math> <math>N=7(b^2+b+1)=a^4</math> for some integer <math>a.</math> Because <math>7\mid a^4</math> and <math>7</math> is prime, <math>a^4 \ge 7^4.</math> Since we want to minimize <math>b,</math> we check to see if <math>a=7</math> works. |
− | When <math> | + | When <math>a=7,</math> <math>b^2+b+1=7^3.</math> Solving this quadratic, <math>b = 18 </math>. |
− | == | + | |
+ | |||
+ | {{Old CanadaMO box|num-b=2|num-a=4|year=1977}} | ||
+ | |||
+ | [[Category:Intermediate Number Theory Problems]] |
Latest revision as of 01:45, 19 August 2012
Problem
is an integer whose representation in base
is
Find the smallest positive integer
for which
is the fourth power of an integer.
Solution
Rewriting in base
for some integer
Because
and
is prime,
Since we want to minimize
we check to see if
works.
When
Solving this quadratic,
.
1977 Canadian MO (Problems) | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • | Followed by Problem 4 |