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Bruce will have both swimming and diving lessons again on June 20th, which is 15 days after the initial date.
To find when Bruce will have both swimming and diving lessons again, we need to determine the least common multiple (LCM) of 3 and 5, which is 15. This means Bruce will have both lessons on a multiple of 15 days. Since he had both lessons on June 5th, the next date will be June 20th (15 days later). Therefore, the correct answer is June 20th.
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