During a brazilian day celebration, dancers perform in groups to honor brazils independance. there are 56 dancers wearing green outfits and 42 dancers wearing yellow outfits. Each group of dancers has the same number of dancers wearing green and the same number of dancers wearing yellow. all of the dancers are in a group. What is the greatest number of groups that can be formed?
To find the greatest number of groups that can be formed, we need to find the greatest common divisor (GCD) of 56 and 42.
The prime factorization of 56 is 2^3 * 7.
The prime factorization of 42 is 2 * 3 * 7.
To find the GCD, we take the smallest exponent of each common prime factor:
2^1 × 7^1 = 14.
Therefore, the greatest number of groups that can be formed is 14.
To find the greatest number of groups that can be formed, we need to find the common factors of 56 (the number of dancers in green outfits) and 42 (the number of dancers in yellow outfits).
The factors of 56 are: 1, 2, 4, 7, 8, 14, 28, 56.
The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42.
The common factors of 56 and 42 are: 1, 2, 7, 14.
Since we want to form groups with an equal number of dancers wearing green and yellow outfits, we can only consider the common factors that evenly divide both 56 and 42.
Therefore, the greatest number of groups that can be formed is 14, as it is the largest common factor of both 56 and 42.
To find the greatest number of groups that can be formed, we need to determine the common factors of 56 (green-dressed dancers) and 42 (yellow-dressed dancers).
Step 1: Find the prime factors of both numbers:
- The prime factors of 56 are 2 x 2 x 2 x 7.
- The prime factors of 42 are 2 x 3 x 7.
Step 2: Identify the common prime factors:
Both 56 and 42 have a common prime factor of 2 and 7.
Step 3: Calculate the product of the common prime factors:
2 x 7 = 14
The number of groups that can be formed is equal to the product of the common prime factors, which is 14. Therefore, the greatest number of groups that can be formed is 14.