I know ETS says that 0 is not a factor of any number except itself but ETS also says that 0/0 is undefined. If x is a factor of y that means y/x gives us an integer. So, 0/0 should give us an integer, 0.
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The definition of a factor doesn’t involve multiplicative inverses. It just matters that you can express 0 as a product of two integer (it’s just a product of two 0's) values thus giving that 0 is a factor of itself. More specifically, 0 | 0 because 0 = 0 \cdot 0.
It is also the case that 0 is not a factor of any other number because 0 \cdot q = 0 regardless of what non-zero integral value q assumes.
To summarize, a factor of x is a y such that for some q we have yq = x. There’s no “division” (multiplicative inverse) involved in the definition, which is a stronger condition in some sense.
No, division by zero is undefined.
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