Question

Josie is making flower arrangements. She has 49 roses and 42 daisies. If Josie wants to make all the arrangements identical and have no flowers left over, what is the greatest number of flower arrangements she can make?

Answer 1

To determine the greatest number of identical flower arrangements that Josie can make without having any flowers left over, we need to find the greatest common divisor (GCD) of the number of roses (49) and daisies (42).

We can find the GCD using the prime factorization method:

1. First, we find the prime factorization of 49:
\[
49 = 7 \times 7 = 7^2
\]

2. Next, we find the prime factorization of 42:
\[
42 = 2 \times 3 \times 7
\]

3. Now, we list the prime factors:
- For 49, the factors are \(7^2\).
- For 42, the factors are \(2^1 \times 3^1 \times 7^1\).

4. The next step is to identify the common prime factors and their lowest powers:
- The only common prime factor is \(7\).
- The lowest power of \(7\) in the two factorizations is \(7^1\).

5. Hence, the GCD is:
\[
7^1 = 7.
\]

Now we will verify this using the Euclidean algorithm:

1. Start with the two numbers \(49\) and \(42\).
- Subtract the smaller number from the larger number:
\[
49 - 42 = 7.
\]

2. Now we have \(42\) and \(7\). We apply the same process:
\[
42 \div 7 = 6 \quad \text{(exact division)}
\]
Since there is no remainder, we conclude that the GCD is \(7\).

Thus, the greatest number of identical flower arrangements Josie can make, with no flowers left over, is:
\[
\boxed{7}.
\]