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IMO Shortlist 1991
Posted: Fri Sep 19, 2014 10:10 am
by mutasimmim
Find the largest power of $1991$ that divides$1990^{1991^{1992}}+1992^{1991^{1990}}$.
Re: IMO Shortlist 1991
Posted: Fri Sep 19, 2014 9:10 pm
by Nirjhor
Apply LTE twice: on \(1990^{1991^{1992}}+1\) and on \(1992^{1991^{1990}}-1\). The answer is \(1991\).
Re: IMO Shortlist 1991
Posted: Fri Sep 19, 2014 10:05 pm
by mutasimmim
Nirjhor, elaborate please. And.. one more thing: $1991$ is not a prime.
Re: IMO Shortlist 1991
Posted: Fri Sep 19, 2014 10:55 pm
by Nirjhor
\(1991=11\times 181\). Work with each of these primes. Solution added to the previous comment.
Re: IMO Shortlist 1991
Posted: Sat Sep 20, 2014 10:26 am
by mutasimmim
$1990^{{1991}^{1992}}+1992^{{1991}^{1990}}=[{1990}^{{1991}^2}]^{{1991}^{1990}}+1992^{{1991}^{1990}}$
Now apply LTE twice.
Re: IMO Shortlist 1991
Posted: Wed Jul 12, 2023 9:15 am
by otis
No power of 1991 is divisible by 199019911992+199219911990.
retro bowl