I wouldn't even think about going after one with anything smaller than a 7mm Magnum, and even that I would find iffy, dependent on the type of bear. Try shooting one with a 5.56x45 or .223 Remington, as the previous poster suggested, and your final epitaph will be read in said bear's bowels.If the game in question is black bear,Then you will be well served by the 308 win,30-06,or a 7mm rem mag.If you are talking a grizzley bear,or alaskan costal brown bear then the demand on the cartridge changes dramatically.I would choose a 338 win mag as the minimum caliber and most alaskan guides choose the 375 H&H or larger because they may have to chase that bear into the willow thickets after being wounded by their clients.
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