To find the greatest amount of identical arrangements, we need to determine the least common multiple (LCM) of the two numbers of balloons.
To find the LCM of 8 and 12, we can follow these steps:
1. List the prime factors of each number: The prime factorization of 8 is 2 x 2 x 2 and the prime factorization of 12 is 2 x 2 x 3.
2. Write down all the prime factors that appear in either number: 2 x 2 x 2 x 3.
3. Multiply all the prime factors: 2 x 2 x 2 x 3 = 24.
Therefore, the LCM of 8 and 12 is 24. This means that the greatest amount of identical arrangements Bryan can make is 24.