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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>
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==Video Solution==
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https://www.youtube.com/watch?v=-rRPkQrmzJw
  
 
==Solution==
 
==Solution==

Latest revision as of 15:43, 28 August 2024

Problem

Determine all positive integers relatively prime to all the terms of the infinite sequence \[a_n=2^n+3^n+6^n -1,\ n\geq 1.\]

Video Solution

https://www.youtube.com/watch?v=-rRPkQrmzJw

Solution

Let $k$ be a positive integer that satisfies the given condition.

For all primes $p>3$, by Fermat's Little Theorem, $n^{p-1} \equiv 1\pmod p$ if $n$ and $p$ are relatively prime. This means that $n^{p-3} \equiv \frac{1}{n^2} \pmod p$. Plugging $n = p-3$ back into the equation, we see that the value$\mod p$ is simply $\frac{2}{9} + \frac{3}{4} + \frac{1}{36} - 1 = 0$. Thus, the expression is divisible by all primes $p>3.$ Since $a_2 = 48 = 2^4 \cdot 3,$ we can conclude that $k$ cannot have any prime divisors. Therefore, our answer is only $1.$

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
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