Verified answer

When one of the numbers is a multiple of the other, for example 33 and 66.

The reason is that all the prime factors of the first number must be included in the second number. Any additional prime factors in the second number will just make it a multiple of the first.

If the first number has factors of:

a * b * c * d

Then the second number has factors of:

a * b * c * d * n

The greatest common factor of both numbers is:

a * b * c * d

which is just the first number.

So one number must just be a multiple of the other.

When one of the numbers goes into the other one evenly. An example is 2 and 4... 4 is the GCF because 2 goes into 4 evenly and 4 goes into 4, so 4 is the GCF

I hope this helps =)

it's pretty obvious that one number has to be a multiple of the other one. you can't get a bigger common factor than the modulus of the smallest of the two numbers, and this occurs if the first one is a multiple of the latter.

When one number is a factor of the other.

This is because the lower number cannot have any factors greater than what it is.

When the number is prime because it's only factors are itself is one.