Question

greatest common factor - applications - item 20055
you are making bouquets to sell. you have 60 roses, 84 daisies, and 24 lilies.
each bouquet will have the same number of each kind of flower. you want to use
all of the flowers. what is the greatest number of bouquets you can make?
4 bouquets
6 bouquets
12 bouquets
24 bouquets

Explanation:

Step1: Find GCD of 60, 84, 24

Prime factorize:
\( 60 = 2^2 \times 3 \times 5 \)
\( 84 = 2^2 \times 3 \times 7 \)
\( 24 = 2^3 \times 3 \)

Step2: Identify common factors

Common prime factors: \( 2^2 \) (since min exponent of 2 is 2) and \( 3^1 \) (min exponent of 3 is 1).

Step3: Calculate GCD

\( \text{GCD} = 2^2 \times 3 = 4 \times 3 = 12 \)? Wait, no—wait, \( 2^2=4 \), \( 4\times3=12 \)? Wait, no, wait: 60, 84, 24. Let's list factors:
Factors of 60: 1,2,3,4,5,6,10,12,15,20,30,60
Factors of 84: 1,2,3,4,6,7,12,14,21,28,42,84
Factors of 24: 1,2,3,4,6,8,12,24
Common factors: 1,2,3,4,6,12. Greatest is 12? Wait, no—wait 24 divided by 12 is 2, 60 divided by 12 is 5, 84 divided by 12 is 7. Yes, so GCD is 12. Wait, but earlier prime factors: 60: 2²×3×5; 84:2²×3×7; 24:2³×3. So common prime factors: 2² (since 60 and 84 have 2², 24 has 2³, so min is 2²) and 3¹ (all have 3¹). So 2²×3=4×3=12. Yes. So the greatest number of bouquets is 12. Wait, but the options: 4,6,12,24. So 12 is an option. Wait, let me check again. 60 ÷12=5, 84÷12=7, 24÷12=2. All integers. So yes, 12.

Answer:

12 bouquets (the option with "12 bouquets")