How do you find the 5th gym badge how do you get to find the 5th gym badge?

First, get the first 4 badges. Then, in Pastoria City, talk to some Galactic grunt in the northeastern corner of the town [in front of the Safari Zone]. Talk to him, start to follow him, and beat your rival. Afterwards, continue following him until you battle him. Once you beat him, go up a little bit, talk to Cynthia, and with the Secretpotion, go to Solaceon Town and north onto Route 210 until you're in front of a group of Psyduck. Talk to them, and then Cynthia appears. She gives you the Old Charm, which you give to her grandma in Celestic Town. Once you go through Route 210, you'll come across Celestic Town. In the middle of the town is another Galactic Grunt. Beat him, and the old woman comes up and takes the Old Charm from you. Go examine the drawings in the cave, and she'll come up again and give you HM03. With this, you can now access the 5th gym in Hearthome City.