What is the greatest common factor of 88 and 16

I always thought that greater common divisor problems can easily be visualized and understood in this way: you know that every number admits an unique prime factorization, i.e. every number is "composed" of certain primes.

Think of these primes as bricks, stacked one on top of each other. So, for example, we have

[tex] 780 = 2^2 \times 3 \times 5 \times 13 [/tex]

and we think that the number 780 is composed by two bricks worth 2, one brick worth 3, one brick worth 5 and one brick worth 13.

If you are fine with this metaphor, the greatest common divisor of two number is simply the bigger stack you can build using only bricks beloning to both numbers.

So, let's see what 88 is made of: its prime factorization is

[tex] 88 = 2^3 \times 11 [/tex]

As for 16, we have

[tex]16 = 2^4[/tex]

So, what do these stacks have in common? We can't use the 11-brick, because 88 doesn't have it. So, we can only use the 2-brick, because both numbers are composed by a certain number of 2-bricks.

Specifically, there are three 2-bricks "inside" 88, and four 2-bricks inside 16. So, if we build a stack with three 2-bricks, that stack will be composed of bricks belonging to both numbers.

We can't add anything else: we can't add the 2-brick 88 doesn't have it, and we can't add the 11-brick because 16 doesn't have it. So, we've actually build the largest stack possible, and the greatest common factor between 88 and 16 is [tex] 2^3 = 8 [/tex]

I hope this alternative visualization helped you out. There obviously is a more rigorous way of solving this kind of problem, so feel free to ask if you want more details on that matter. After all, the substance is all here already, you only have to give a proper name to things, instead of talking about "bricks" and "stacks", but I fell I may make things easier for you, offering some kind of visualization