Difference between revisions of "1977 Canadian MO Problems/Problem 3"
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== Solution == | == Solution == | ||
| − | 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 \ge 7^4.</math> Since we want to minimize <math>b,</math> we check to see if <math>a=7 | + | 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>a=7 | + | 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}} | {{Old CanadaMO box|num-b=2|num-a=4|year=1977}} | ||
| − | [[Category: | + | [[Category:Intermediate Number Theory Problems]] |
Latest revision as of 00:45, 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 |
