Some Notes About the Problem
This question is testing your ability to take information from a word prompt and figure out what the underlying math problem is. If you can do that, the math problem itself actually isn't that difficult in this case (and in many cases).
In my opinion, this question isn't particularly well-written, since the constraints of the problem aren't defined well enough to permit only 1 or 2 correct answers. If the wording of the problem is confusing to you, I wouldn't judge you very harshly. As it is written, there are many, many correct answers. Let's try to figure out what the question is probably intending to ask.
Firstly, the condition "put an equal number of roses and carnations into each centerpiece" has a partially arbitrary wording. What I mean is that it can mean more than one different thing depending on how you interpret it.
Does it mean this?
-In every centerpiece, the number of roses is equal to the number of carnations. (1 rose and 1 carnation, or 2 roses and 2 carnations, or 3 roses and 3 carnations...)
Or does it mean this?
-Every centerpiece has the same number of roses, and also every centerpiece has the same number of carnations. (Roses are divided evenly among each centerpiece, and carnations are divided evenly among each centerpiece.)
Since the first meaning is trivial to solve for no matter how many constraints we have, let's focus on the second, more interesting one. The second one also makes sense because usually centerpieces on different tables are made to look the same, so we should assume each centerpiece is made identically.
Next, I would like to point out that the problem never states that all of the carnations and all of the roses must be used. If we are trying to find out "the number of centerpieces the florist can make", which can reasonably mean "what is the BIGGEST number the florist can make", then the answer is 70, because each centerpiece can be made with exactly 1 carnation and 0 roses. ...But this is sort of a smart-mouthed answer, isn't it? I think it's reasonable to assume that we are supposed to try to use all the flowers, and each centerpiece must have at least 1 of each kind.
So, let's add the following constraints based on the above reasoning:
-Make as many centerpieces as possible.
-Use all of the flowers.
-Use both carnations and roses in each centerpiece.
Solving the Problem
If we want to divide a number evenly into parts, we are just using that number's factors.
In this case, we are dividing two numbers evenly into the same number of parts. That means that we are finding a common factor. And since we are looking for the greatest possible number of parts (the biggest number of centerpieces), that means we are looking for the greatest common factor (GCF).
All we need to do is find the factors of both numbers, and figure out which factors appear for both. Then we just pick the biggest one from those. Let's try it:
Factors of 56:
1
2
4
7
8
14
28
56
Factors of 70:
1
2
5
7
10
14
35
70
The common factors (numbers that are factors of both 70 and 56) are:
1
2
7
14
And of course, the biggest one of the common factors is 14, therefore the GCF of 56 and 70 is 14.
Remember, the GCF of two numbers really just means "the biggest number that can evenly divide both numbers", which is exactly what our problem is asking for-- the biggest number of centerpieces we can make by evenly dividing 56 roses and 70 carnations into the same number of parts is 14 centerpieces.
To check our work, let's divide both numbers by 14:
56 roses / 14 centerpieces = 4 roses / centerpiece (say "4 roses per centerpiece")
70 carnations / 14 centerpieces = 5 carnations / centerpiece ("5 carnations per centerpiece)
They both divide evenly! And we know we can't make any more centerpieces by using less flowers in each, because our factors above show there are no common factors bigger than 14; we are done.
