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I think this class is already implemented in sage:
sage: a,b
(2 * 3, 3 * 5)
sage: a*b
2 * 3^2 * 5
sage: a=factor(6); a
2 * 3
sage: b=factor(15); b
3 * 5
sage: a*b
2 * 3^2 * 5
sage: type(a*b)
<class 'sage.structure.factorization.Factorization'>
sage: type(a)
<class 'sage.structure.factorization_integer.IntegerFactorization'>
A good start would be to compare your code with what is implemented in sage already. It might give some hints.
