Q: The gcf of any two even numbers is 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.

Yes.

The GCF of two even numbers is always even.

True.

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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.

Yes.

The GCF of two even numbers is always even.

No

Yes.

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

True.

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.

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

Yes and yes.

Yes, that is true.