Question

scenario 3 next, luis works on a dance which will mix dancers from two different dance schools. he wants to divide the two schools of dancers into smaller teams with the same number of dancers from each school on each team. one school has 36 dancers and the other school has 54 dancers. what are the team sizes he can use? what is the greatest team size that luis can choose? generalize how is scenario 3 different from scenarios 1 and 2?

Explanation:

Part 1: Team Sizes Luis Can Use

Step 1: Find Factors of 36

To find the factors of 36, we list all numbers that divide 36 without leaving a remainder:
$1, 2, 3, 4, 6, 9, 12, 18, 36$

Step 2: Find Factors of 54

Similarly, find factors of 54:
$1, 2, 3, 6, 9, 18, 27, 54$

Step 3: Identify Common Factors

The common factors of 36 and 54 are the numbers present in both lists:
$1, 2, 3, 6, 9, 18$

Part 2: Greatest Team Size (Greatest Common Factor)

Step 1: Prime Factorization of 36

Break down 36 into prime factors:
$36 = 2 \times 2 \times 3 \times 3$

Step 2: Prime Factorization of 54

Break down 54 into prime factors:
$54 = 2 \times 3 \times 3 \times 3$

Step 3: Identify Common Prime Factors

The common prime factors are $2, 3, 3$ (the product of the lowest powers of common primes).

Step 4: Calculate GCF

Multiply the common prime factors:
$2 \times 3 \times 3 = 18$

Part 3: Generalize (Comparison to Other Scenarios)

Assuming Scenarios 1 and 2 involved a single group (or a different context, e.g., distributing items to one group), Scenario 3 involves two distinct groups (dancers from two schools). Instead of finding factors for one number, we find common factors (or GCF) of two numbers to ensure equal division from both groups. This requires analyzing divisibility for multiple quantities, unlike scenarios focused on a single quantity.

Answer:

s:

• Team sizes Luis can use: $1, 2, 3, 6, 9, 18$
• Greatest team size: $18$
• Generalization: Scenario 3 involves two groups (dancers from two schools), so we find common factors/GCF of two numbers (36 and 54) to ensure equal division from both, unlike scenarios with one group.