The drama club meets in the school auditorium every 3 days, and the choir meets there every 12 days. If the groups are both meeting in the auditorium today, then how many days from now will they next have to share the auditorium?

Answer 1

To find out when the drama club and the choir will next meet in the auditorium, we need to determine the least common multiple (LCM) of the two meeting schedules: every 3 days for the drama club and every 12 days for the choir.

We start by finding the prime factorizations of 3 and 12:

• 3 is a prime number, so its prime factorization is \( 3^1 \).
• 12 can be factored as \( 12 = 3 \times 4 = 3^1 \times 2^2 \).

To find the LCM, we take the highest power of each prime number that appears in the factorizations:

• For the prime number 3, the highest power in the factorizations is \( 3^1 \).
• For the prime number 2, the highest power in the factorizations is \( 2^2 \).

Thus, the LCM can be calculated as follows:

\[ \text{LCM} = 3^1 \times 2^2 = 3 \times 4 = 12. \]

This means that the drama club and the choir will both meet again in the auditorium in 12 days.

Therefore, the number of days from now until they next share the auditorium is \( \boxed{12} \).