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?
36 bouquets
6 bouquets
3 bouquets
9 bouquets
To find the largest number of bouquets the florist can make without having any flowers left over, we need to find the common factor of 36, 27, and 18.
The prime factorization of 36 is 2^2 * 3^2.
The prime factorization of 27 is 3^3.
The prime factorization of 18 is 2 * 3^2.
Based on the prime factorization, we can see that the common factor of 36, 27, and 18 is 3^2 = 9.
Therefore, the florist can make 9 bouquets without having any flowers left over.
To find the largest number of bouquets the florist can make without having any flowers left over, we need to determine the greatest common divisor (GCD) of the given numbers: 36, 27, and 18.
The common factors of 36, 27, and 18 are 1, 2, 3, 6, and 9. The largest number that divides all three numbers evenly is 9.
Therefore, the florist can make 9 bouquets without having any flowers left over.
To find the largest number of bouquets that the florist can make without having any flowers left over, we need to find the greatest common divisor (GCD) of the number of each type of flower.
The GCD is the largest number that divides all the numbers evenly. In this case, we have 36 roses, 27 tulips, and 18 carnations.
To find the GCD, we can use the prime factorization method.
For the roses (36), we can prime factorize it as 2 x 2 x 3 x 3.
For the tulips (27), we can prime factorize it as 3 x 3 x 3.
For the carnations (18), we can prime factorize it as 2 x 3 x 3.
Now, we look at the common prime factors among the three numbers: 2, 3, and 3.
To find the GCD, we take the smallest exponent for each common prime factor:
2 x 3 x 3 = 18
Therefore, the GCD of 36, 27, and 18 is 18.
Since the GCD is 18, it means that the florist can create bouquets with 18 flowers of each type without any flowers left over.
To find the number of bouquets, we divide the number of each type of flower by the GCD:
Number of roses / GCD = 36 / 18 = 2 bouquets
Number of tulips / GCD = 27 / 18 = 1.5 bouquets
Number of carnations / GCD = 18 / 18 = 1 bouquet
Since we need to have an equal number of each type of flower in each bouquet, we take the smallest whole number as the number of bouquets that can be made:
2 bouquets
Therefore, the largest number of bouquets the florist can make without having any flowers left over is 2 bouquets.