Question

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.
How many roses will be in each bouquet?

Answer 1

To determine how many roses will be in each bouquet, we first need to find the greatest common divisor (GCD) of the quantities of the flowers: 36 roses, 27 tulips, and 18 carnations. The GCD will tell us how many bouquets can be made with an equal number of each flower.

1. **Finding the prime factorization of each quantity:**
- For 36:
\[
36 = 2^2 \times 3^2
\]
- For 27:
\[
27 = 3^3
\]
- For 18:
\[
18 = 2^1 \times 3^2
\]

2. **Identifying the common factors:**
- The prime factors involved are \(2\) and \(3\).
- For \(2\), the minimum exponent in the factorizations is \(0\) (because \(27\) does not include \(2\)).
- For \(3\), the minimum exponent is \(2\) (from both \(36\) and \(18\)) and occurs in all three.

3. **Calculating the GCD:**
\[
\text{GCD} = 2^0 \times 3^2 = 1 \times 9 = 9
\]

4. **Determining the number of bouquets:**
Since the GCD is \(9\), this means we can create \(9\) bouquets.

5. **Finding out how many roses will be in each bouquet:**
Each bouquet will contain:
\[
\frac{36 \text{ roses}}{9 \text{ bouquets}} = 4 \text{ roses per bouquet}
\]

Thus, the number of roses in each bouquet is \(\boxed{4}\).