Final answer:
The first caller to win both a free concert ticket and a free link ride is the 60th caller, as this is the Least Common Multiple (LCM) of 12 and 15.
Step-by-step explanation:
The question inquires which caller will be the first one to win both a free concert ticket and a free link ride, given that every 12th caller receives a ticket and every 15th caller receives a link ride.
To find out which caller will be the first to win both, we need to determine the Least Common Multiple (LCM) of 12 and 15, which is the smallest number that both 12 and 15 will divide into without leaving a remainder.
Using prime factorization:
12 = 2² × 3
15 = 3 × 5
The LCM of 12 and 15 is the product of the highest power of each prime factor present in the factorization of both numbers, which is:
LCM(12, 15) = 2² × 3 × 5
= 4 × 3 × 5
= 60
Therefore, the first caller to win both prizes will be the 60th caller, as this is the first number that is both a multiple of 12 and a multiple of 15.