Since any odd number can be a factor, there is an infinite amount of them.
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There are 11 numbers between 1 and 130 that have an odd number of factors: 1,4,9,16,25,36,49,64,81,100,121.
Yes, the factors of all odd numbers are odd numbers.
Square numbers have an odd number of factors, but beyond that, any odd number can be a factor.
Just the one which is 7
Square numbers have an odd number of factors.