Final answer:
The greatest number of identical arrangements without leftovers can be found by calculating the greatest common divisor (GCD) of 54 and 42, which is 6. This results in 6 identical arrangements consisting of 9 roses and 7 tulips each.
Step-by-step explanation:
To find the greatest number of identical arrangements that can be made with 54 roses and 42 tulips with no leftovers, we need to calculate the greatest common divisor (GCD) of the two quantities. The GCD is the largest number that divides both numbers without any remainder. To find the GCD, we can use the Euclidean algorithm or simply list the factors of both numbers and find the largest one they have in common. In this case, the GCD of 54 and 42 is 6. Therefore, we can create 6 identical arrangements, each containing 9 roses (54/6) and 7 tulips (42/6), with no flowers left over.