All nonzero numbers have factors. Some factors are even numbers, some factors are odd numbers.
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Not only can they, but they must. An odd number cannot have an even factor. So none of the factors of either of the two odd numbers can be even and so the GCF is made up of only odd factors and so it must be odd.
At least one of the factors of an even number must be even, because the product of odd factors is always odd.
No. Odd numbers only have odd factors. They could not have an even factor in common.
The GCF of an odd and even number is never even.
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