A florist has 36 roses, 27 tulips, and 18 carnations she must use to create bouquets. Each bouquet must have an equal number of each type of flower as the other bouquets. What is the largest number of bouquets she can make without having any flowers left over? (1 point) Responses 36 bouquets 36 bouquets 6 bouquets 6 bouquets 3 bouquets 3 bouquets 9 bouquets
To find the largest number of bouquets she can make without having any flowers left over, we need to find the greatest common divisor (GCD) of 36, 27, and 18.
The prime factorization of 36 is 2^2 * 3^2 * 1.
The prime factorization of 27 is 3^3.
The prime factorization of 18 is 2 * 3^2.
To find the GCD, we take the smallest exponent for each prime factor.
The smallest exponent for 2 is 2.
The smallest exponent for 3 is 2.
So, the GCD is 2^2 * 3^2 = 4 * 9 = 36.
Therefore, the largest number of bouquets she can make without having any flowers left over is 36 bouquets.
To find the largest number of bouquets the florist can make without any flowers left over, we need to find the greatest common divisor (GCD) of the quantities of each type of flower.
The GCD of 36, 27, and 18 is 9.
So, the largest number of bouquets she can make without having any flowers left over is 9 bouquets.
To find the largest number of bouquets the florist can make without having any flowers left over, we need to find the greatest common divisor (GCD) of the numbers of roses, tulips, and carnations.
The GCD represents the largest number that divides evenly into all three numbers. In this case, we have 36 roses, 27 tulips, and 18 carnations.
To find the GCD, we can use the method of prime factorization:
1. Prime factorize each of the numbers:
- 36 = 2^2 * 3^2
- 27 = 3^3
- 18 = 2 * 3^2
2. Identify the common prime factors:
- The common prime factors are 2^2 and 3^2.
3. Multiply the common prime factors to find the GCD:
- GCD = 2^2 * 3^2 = 4 * 9 = 36
So, the GCD of 36 roses, 27 tulips, and 18 carnations is 36.
To determine the largest number of bouquets she can make without having any flowers left over, we divide each number by the GCD:
- Number of roses: 36 / 36 = 1 bouquet
- Number of tulips: 27 / 36 = 3/4 of a bouquet (not a whole number)
- Number of carnations: 18 / 36 = 1/2 of a bouquet (not a whole number)
Since we need whole numbers for each type of flower in each bouquet, the limiting factor is the number of tulips and carnations. Therefore, we need to consider the lowest common multiple (LCM) of 4 and 2 in order to create equal numbers.
The LCM of 4 and 2 is 4, so we can group 4 bouquets together to ensure an equal number of flowers in each bouquet.
Therefore, the largest number of bouquets she can make without having any flowers left over is 1 bouquet of roses, 4 bouquets of tulips (4 * 3 = 12 tulips), and 2 bouquets of carnations (2 * 6 = 12 carnations).
So, the correct answer is 1 bouquet.