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Q: Is the GCF of any two even numbers always even?

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The GCF of any two odd numbers is always odd. The GCF of any two even numbers is always even. The GCF of an odd and an even number is always odd.

The GCF of any two odd numbers is always odd because odd numbers don't have any even factors. The GCF of any two even numbers is always even because even numbers are divisible by two and any common factors would have at least one two in common. The GCF of an even and an odd number is odd because odd numbers don't have any even factors.

No

That is correct.

No, always odd.

The GCF of any set of odd numbers is odd because odd numbers don't have any even factors.

All even numbers have two as a factor. Any two even numbers have (at the least) two as a common factor. That means the GCF of any two even numbers will have two as a factor. Any number that has two as a factor is even.

Yes.

All even numbers have two as a factor. Since two will be a common factor of any set of even numbers, the GCF will also have two as a factor, making it even.

True.

No. Since odd numbers don't have any even factors, the GCF of odd and even must be odd.

Yes.

No, always odd. Since odd numbers don't have any even factors, they can't have any in common.

Yes.

True. Even numbers are even because they are multiples of 2. That means that any two even numbers will have at least 2 as a common factor. Since that number (the GCF) will have 2 as a factor, it will be a multiple of 2: an even number.

Yes, that is true.

Yes, that's true.

By definition, every even number has 2 as a factor. That means that, at a minimum, every set of even numbers will have a common factor of 2. The GCF could be higher, of course, but that common factor of 2 will always be there, making the numbers even.

If x is an odd number that is a common factor of 2 even numbers, so is 2x. So the answer is yes.

No. The GCF of 6 and 7 is 1.

Yes because if it is even then all of it's factors will be even. It is fairly different if it's an odd number. But I will not get into that.

Because odd numbers don't have any even factors so they can't have any in common.

Any consecutive even numbers have a GCF of 2.

Any consecutive even numbers.

No, the GCF of any two numbers may be odd or even. For instance:The GCF of 2 and 4 is 2The GCF of 3 and 6 is 3The GCF of 14 and 21 is 7.