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Suppose A and B are two even numbers. Then, by definition,

A = 2*P for some integer P, and

B = 2*Q for some integer Q

So 2 is a factor of both A and B, that is it is a common factor.

Now, any common factor of A and B is, by definition, a factor of their HCF. That is, 2 is a factor of the HCF of A and B: in other words, the HCF is even.

Q: Why is the gcf of two even numbers are always even?

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Yes.

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.

The sum of two even numbers is always an even number.

Always.

There are infinitely many pair of numbers. Take any two numbers that are coprime. Double both of them. Their GCF is 2.

Related questions

The GCF of two even numbers is always even.

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.

No. For example, the GCF of 8 and 12 is 4. The GCF of 12 and 18 is 6. The Greatest Common Factor of two even numbers will always be even, but it will not always be 2.

Yes.

Yes.

That is correct.

Yes.

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

No not always as for example the GCF of 7 and 21 is 7

No. The GCF of 8 and 12 is 4.

No. The GCF of 4 and 8 is 4.

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