No. Consider 15: 1,3,5,15
Every positive whole number has an even number of factors,
unless the number is a perfect square.
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Square numbers have an odd number of factors.
All square numbers have an odd number of factors.
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.
Perfect squares are numbers with an odd number of factors.
Square numbers have an odd number of factors, but beyond that, any odd number can be a factor.