Question

greatest common factor - applications - item 31130
a florist has 56 roses and 70 carnations to make table centerpieces. he wants to put an equal number of roses and carnations into each centerpiece.
use the drop - down menus to complete the statements below about the number of centerpieces the florist can make if he uses all the flowers.
the greatest number of centerpieces the florist can make is. each of theses centerpieces will have roses and carnations.
if he made more centerpieces, .
he could make centerpieces, but this is not the greatest number he could make.

Explanation:

Step1: Find GCF of 56 and 70

Factorize 56: \(56 = 2\times2\times2\times7\)
Factorize 70: \(70 = 2\times5\times7\)
GCF is \(2\times7 = 14\). So max centerpieces: 14.

Step2: Calculate flowers per centerpiece

Roses per centerpiece: \(\frac{56}{14} = 4\)
Carnations per centerpiece: \(\frac{70}{14} = 5\)

Step3: Analyze more centerpieces

If more than 14, say 15, \(\frac{56}{15}\) or \(\frac{70}{15}\) not integer. So can't use all flowers.

Step4: Find non - max centerpieces

Factors of GCF (14) or common factors. Factors of 14: 1,2,7,14. Non - max: 1,2,7. E.g., 7: \(\frac{56}{7}=8\), \(\frac{70}{7}=10\) (integers).

Answer:

The greatest number of centerpieces the florist can make is 14. Each of these centerpieces will have 4 roses and 5 carnations.
If he made more centerpieces, he could not use all the flowers (or similar reasoning about non - integer division).
He could make 7 (or 1, 2) centerpieces, but this is not the greatest number he could make.