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?

Question

UpStudy AI · Accepted Answer

Let the number of bouquets be $$ x $$. Since each bouquet has the same number of roses, daisies, and lilies, we must have:

$$
\frac{60}{x}, \quad \frac{84}{x}, \quad \frac{24}{x} \quad \text{all integers.}
$$

This is possible if and only if $$ x $$ is a divisor of 60, 84, and 24. We are asked for the greatest number of bouquets, which means we need to find the greatest common divisor (gcd) of 60, 84, and 24.

**Step 1. Find $$ \gcd(60, 84) $$:**

$$
60 = 2^2 \times 3 \times 5, \quad 84 = 2^2 \times 3 \times 7.
$$

The common factors have:

$$
\gcd(60, 84) = 2^2 \times 3 = 4 \times 3 = 12.
$$

**Step 2. Find $$ \gcd(12, 24) $$:**

$$
12 = 2^2 \times 3, \quad 24 = 2^3 \times 3.
$$

Taking the minimum powers of the common primes:

$$
\gcd(12, 24) = 2^2 \times 3 = 4 \times 3 = 12.
$$

Thus,

$$
\gcd(60, 84, 24) = 12.
$$

The greatest number of bouquets you can make is therefore $$ \boxed{12} $$.
You can make 12 bouquets.

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?

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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?

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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?