These posts pretty much covered the subject, so there's not too much advice I can add about choices. All of those calibers would be adequate for deer, elk and any other Rocky Mountain game. Your $500 limit would be quite a stretch for a scope AND rifle, at least for anything that was new. You might have to consider a used rifle and scope. Nothing wrong with that as most rifles outlive their original owners. Bill Ruger once said that gun manufacturers were in competition with themselves because the guns they made lasted so long. Here's a tip on getting a used scope: some manufacturers guarantee their scopes forever - even extending this to non-original owners. Leupold and Burris are two companies that do this, and they are both excellent quality. I favor the Leupold because of the overall quality and long eye relief. Good luck.
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