Yes once per game.
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I would like to add that if you mean the one with Buena on the second floor, you can do it (I'm pretty sure) once a day. If you go to your Pokegear and listen to various Channels, you might be able to listen to Buena's Password. During the radio show will tell you the answer to the quiz.
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You must answer the quiz in the Radio Tower correctly and the gym leader will go back to the gym.
You can get to the radio tower in HeartGold by going to Goldenrod City and if you go there there is a daily drawing corner and a quiz to get the radio card. If there is someine blocking you, then go to the underground shop, get the Team Rocket uniform, and they shall let you in.
Whitney is in the radio tower. She is trying to get the radio card. You have to beat the quiz and she will return to her gym
When you get to the Radio Tower, you go in. The next thing you do is, you go all the way down to the end of the counter and talk to the lady there. The lady is next to the man in the middle counter. The lady will ask if you want to take the quiz. By the way, she is on the right of the counter, just to let you know.
first you have to be able to go to blackthorn city - then you fly to goldenrod city and visit the radio tower - team rocket will be there you have to defeat them and get the director (afterwards) - then you will need to fly to blackthorn city, you will then be able to go into the gym beat the gym and then go into dragons den answer the guys quiz then Clair will appear then go out and she will come running and give you the badge. DONE!