Question

A florist has 36 roses and 27 tulips, and 18 carnations she must use to create a 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

Answer 1

To solve this problem, we need to find the greatest common divisor (GCD) of the number of each type of flower available. The florist has:

- 36 roses
- 27 tulips
- 18 carnations

We start by finding the GCD of these three numbers.

1. **Finding the prime factorization of each number**:

- The prime factorization of 36 is \(2^2 \times 3^2\).
- The prime factorization of 27 is \(3^3\).
- The prime factorization of 18 is \(2^1 \times 3^2\).

2. **Identifying the common prime factors**:

The only prime factor that appears in all three factorizations is 3.

3. **Finding the minimum exponent**:

- For 3:
- In 36, the exponent is 2.
- In 27, the exponent is 3.
- In 18, the exponent is 2.
- The minimum exponent for 3 is \(2\).

Since the number 2 from the prime factorization of the flowers does not appear in all three numbers, we only consider the factor \(3^2\).

Thus, the GCD of 36, 27, and 18 is:

\[
GCD = 3^2 = 9
\]

4. **Calculation of the largest number of bouquets**:

The largest number of bouquets the florist can create, where each bouquet contains an equal number of each type of flower without any left over, is therefore:

\[
\boxed{9}
\]