Answer to Question #93564 in Combinatorics | Number Theory for Emely

Question

A pro baseball team is having a promotion in which every tenth fan that enters the stadium gets a free hat and every 12th person gets a free t shirt. How many fans will come into the stadium before a fan receives a hat and a t shirt?

Answer 1

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:

$10=2\cdot 5$

$12=2\cdot2\cdot3$.

LCM is the product of maximum occurrences of each prime factor in the given numbers.

$LCM(10,12)=2\cdot2\cdot3\cdot5=60$

Hence, 59 fans will come into the stadium before the 60th fan receives a hat and a T-shirt.