If a fan receives a hat and a T-shirt, then his number is divisible by 10 and 12. Therefore, this number is the least common multiple of 10 and 12.
Prime factorizations of 10 and 12 are:
.
LCM is the product of maximum occurrences of each prime factor in the given numbers.
Hence, 59 fans will come into the stadium before the 60th fan receives a hat and a T-shirt.
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