The music teacher can split the class into groups of 8 or 12. each group will have the same amount of students. what is the smallest number of people that can be at music class?

Since they can be groups of 8 or 12 the minimum number of students will be equal to the least common multiple of 8 and 12.  

The least common multiple is the product of the highest occurring primes from the two number prime factorization...

8=2*2*2 and 12=2*2*3  so the least common multiple will be 2*2*2*3=24

So there must be at least 24 people in the class...