The florist makes the greatest number of identical arrangements with the carnations and asters. She has 72 carnations and 42 asters. How can she decide how many carnations to place in each arrangement?

Question

Subtract the number of asters from the number of carnations.
Divide the number of carnations by the GCF of 72 and 42.
Find the least common multiple of 72 and 42.
Find the greatest common factor of 72 and 42.

Answer 1

To determine how many carnations to place in each arrangement, the florist should find the greatest common factor (GCF) of the number of carnations and asters. The GCF will help her decide how to evenly distribute the flowers into identical arrangements.

Here are the steps to find the GCF of 72 and 42:

1. Prime Factorization:

   • 72 can be factored into \(2^3 \times 3^2\).
   • 42 can be factored into \(2^1 \times 3^1 \times 7^1\).

2. Identify the Common Factors:

   • The common prime factors are \(2\) and \(3\).
   • For \(2\), the minimum exponent is \(1\) (from 42).
   • For \(3\), the minimum exponent is \(1\) (from 42).

3. Calculate the GCF:

   • GCF = \(2^1 \times 3^1 = 2 \times 3 = 6\).

Hence, the greatest common factor of 72 and 42 is 6.

This means the florist can make arrangements with 6 carnations in each one if she wants to evenly distribute her flowers into identical arrangements.

So the correct answer is: Find the greatest common factor of 72 and 42.