## JBMO-2012-4

For discussing Olympiad Level Number Theory problems
Tahmid Hasan
Posts: 665
Joined: Thu Dec 09, 2010 5:34 pm

### JBMO-2012-4

Find all positive integers $x,y,z$ and $t$ such that $2^{x}3^{y}+5^{z}=7^{t}$
বড় ভালবাসি তোমায়,মা

Posts: 1016
Joined: Tue Nov 22, 2011 7:49 pm
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### Re: JBMO-2012-4

This problem offers you two options: either do huge calculations or use Zsigmondy's theorem. Though none of them is attractive, but, at my point of view, the first option is much more honorable.
My Solution:
Case:1 $x=1$
Case: 2 $x>1$
Case: $2.1$ $q=0$
Case: $2.2$ $q=y$
Here also two cases:
Case: $2.2.1$ $p=1$
Case: $2.2.2$ $x-p=1$
solution(s):
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