What is the smallest number of people that can be broken up both into 15 groups of equal membership and into 48 groups of equal membership?
Let $x$ be the smallest number of people that can be broken up into 15 groups of equal membership and into 48 groups of equal membership.  This means $x$ must be a multiple of both 15 and 48.  The smallest such number is the least common multiple of 15 and 48. $15=3 \cdot 5$ and $48=2^4 \cdot 3$. Thus, any multiple of 15 and 48 must have a factor of 2 raised to at least the fourth power, a factor of 3 raised to at least the first power, and a factor of 5 raised to at least the first power. Thus, the least such multiple is $2^4 \cdot 3\cdot 5 = \boxed{240}$.
240
Level 3
Prealgebra
Problem: What is the smallest number of people that can be broken up both into 15 groups of equal membership and into 48 groups of equal membership?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the smallest number of people that can be divided into both 15 groups and 48 groups, we need to determine the least common multiple (LCM) of the two numbers, 15 and 48. 

First, we factor each number into its prime components. The number 15 can be factored as 3 and 5, so its prime factorization is 3^1 * 5^1. The number 48 can be factored as 2^4 and 3^1, giving it a prime factorization of 2^4 * 3^1.

Next, to find the LCM, we take the highest power of each prime that appears in the factorizations. For the prime number 2, the highest power is 2^4 from 48. For the prime number 3, the highest power is 3^1, which appears in both factorizations. For the prime number 5, the highest power is 5^1 from 15.

Now, we multiply these highest powers together to find the LCM:
LCM = 2^4 * 3^1 * 5^1.

Calculating this step-by-step:
1. Calculate 2^4 = 16.
2. Calculate 3^1 = 3.
3. Calculate 5^1 = 5.
4. Now multiply these results: 16 * 3 = 48.
5. Finally, multiply 48 by 5: 48 * 5 = 240.

Thus, the smallest number of people that can be divided into both 15 groups and 48 groups is 240.<|END_OF_LATENT|> The final answer is $240$.