Can a number such as 10 have infinite factors if we include multiplying two rational numbers or a rational number and integer or any other combinations? Google says factors of 10 are 1×10, 2×5 and the number of factors is finite but isn't 2.5×4 also a factor of 10 and infinite more?.
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1$\begingroup$ A "factor" of an element in a ring is not a proper factor if it is a unit. So in $\Bbb Z$ we would consider the units, which are $\pm 1$, as trivial factors. However in $\Bbb Q$ all nonzero numbers are units. So if you talk of $\frac{5}{2}$ as a factor in $\Bbb Q$, it is not a proper factor. $\endgroup$– Dietrich BurdeCommented May 20, 2022 at 13:48
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$\begingroup$ You observe that if we allow rationals then there are infinitely many factors, and indeed every nonzero rational number is a factor of every other nonzero rational number. So this is a rather useless definition, since if everything but zero is a factor of every number then there’s simply no point to discussing “factors” anymore. A slightly more abstract take: the integers are only a ring, whereas the rationals are a field. Rings don’t have division, so factorization is not trivial and is worth studying. In a field, we have division, and this additional structure makes factorization trivial. $\endgroup$– Kenanski BowspleefiCommented May 20, 2022 at 14:01
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It depends more on what you define a factor to be.
In the context of number theory it might suit you better to define a factor as an element in the ring which divides your number or if you work in a UFD a factor is a prime (or combination of such) which appears in the unique factorization. In this case, speaking of the ring $\mathbb(Z)$ you obviously can only have a finite amount of factors for any integer such as $10$.
If you include rational numbers in your definition of factor it obviously looks a bit different. You can find infinite factorizations into two rational numbers for any given rational.
For example with 10 you get $\frac{1}{2}*20, \frac{1}{3}*30, \frac{1}{4}*40$ and so on. And theres even infinite more.
So your question more or less boils down to what you take as a definition for the word "factor". In different fields of maths it can make sense to define this term differently
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$\begingroup$ Also worth adding that if $n$ is an integer, then when we are almost always speaking about the "factors" of $n$, we always mean "the integer factors of $n$", or in other words "the factors of $n$ in the ring $\mathbb Z$". $\endgroup$– 5xumCommented May 20, 2022 at 14:12