he does not give you a gym badge, but you fight him at the end of Mt. ortar, rock climb is required. hope that helps...
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The last badge is off the coast of cinnabar it is the red dot in the left corner of the map
You need to defeat Koga in Fuchsia City to get the 5th badge, the Soul Badge.
U win
the 13th badge you get in soul silver is in fucshia city, from janine she is a poison type expert, as she is koga-of-the-elite-four's daughter.
Try and get the 5th one lol.