Assuming I understand this correctly. Given two numbers x and y, there always is some gcf and you can rewrite both as a multiple of that gcf and the remaining parts k1 and k2 are relatively prime (you can think of them as a product of primes themselves if that makes it easier).
i..e x = k1 * gcf and y = k2 * gcf
Then when subtract the two the distributive law says the difference will still be a multiple of the gcf.
i.e. assume x is greater than y
x - y = k1 * gcf - k2 * gcf = (k1 - k2) * gcf
So this difference definitely shares the original gcf as a common factor with both original numbers x and y. And the gcf must still be the greatest common factor for the inverse reason. If the other part of the product k1 - k2 contained another factor > 1 with either number then this factor would have to have been in both originals via the distributive law and we'd have a contradiction with the original gcf being the greatest factor.
This looks a bit formal on rereading so I hope this helps.
This makes perfect sense. But for some reason, in my mind, that English explanation doesn't communicate this. The and their difference seems....vague? I don't know.
Also, this seems like a weird way to explain it in words. The mathematical notation is very "duh" too me, but this wording? Huh. I'm thinking a little too hard.
"The GCF of any 2 numbers is the same as the GCF of either number and their difference."
I think that more than the concept, the wording of this, especially the red and blue part is a little confusing. Maybe that is what you're daughter is struggling with OP.
re the red: There can ONLY be a greatest common factor if there are two or more. For some reason, in my mind the word "either" would instinctively translated to one of the numbers OR the other number being factored.
12 does not have a GCF, but 12 and 16 do have a GCF.
re: the blue, I was confused because "their" seemed like it was talking about the GCF and I was all Because the GCF is just one number, so GCF-GCF would 0.
I get it now, but on the first several readings this seems like a bad English "translation" of what they are getting at. Since she's struggling with the definition that uses multiplication-terms, I would walk-through this with your daughter and find the equivalent definition that uses divisibility-terms.
That might help her see the concept without the English clouding up her math.
Basically it's saying "when you have the numbers. a, b, then assuming that a > b, you can have: a, b and (a-b) as your 3 numbers.
If TWO of those numbers are divisible by n, then so is the third number. Guaranteed.
OP, maybe your daughter just needs help unpacking the verbiage the way that I did?