Difference between revisions of "2005 IMO Problems/Problem 4"
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Determine all positive integers relatively prime to all the terms of the infinite sequence <cmath> a_n=2^n+3^n+6^n -1,\ n\geq 1. </cmath> | Determine all positive integers relatively prime to all the terms of the infinite sequence <cmath> a_n=2^n+3^n+6^n -1,\ n\geq 1. </cmath> | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://www.youtube.com/watch?v=-rRPkQrmzJw | ||
==Solution== | ==Solution== | ||
Latest revision as of 15:43, 28 August 2024
Contents
Problem
Determine all positive integers relatively prime to all the terms of the infinite sequence ![\[a_n=2^n+3^n+6^n -1,\ n\geq 1.\]](http://latex.artofproblemsolving.com/a/3/f/a3f3c15ee9d92c2eb2747f800d908cfa7aa6a08b.png)
Video Solution
https://www.youtube.com/watch?v=-rRPkQrmzJw
Solution
Let 
be a positive integer that satisfies the given condition.
For all primes 
, by Fermat's Little Theorem, 
if 
and 
are relatively prime. This means that 
. Plugging 
back into the equation, we see that the value
is simply 
. Thus, the expression is divisible by all primes 
Since 
we can conclude that cannot have any prime divisors. Therefore, our answer is only 
See Also
| 2005 IMO (Problems) • Resources | ||
| Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
| All IMO Problems and Solutions | ||
