Is there any trick to find the number of divisors of any number? For e.g., a quick way to tell the number of divisors of 987655432 (chosen randomly)?
EDIT: And it has to be done without prime factorization.
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Consider the natural number $987655432$. Using prime factorization (and hating our lives while using it, lol) we see that $987655432=2^3\cdot 1033\cdot 119513$. By the Fundamental Theorem of Arithmetic, each factor is of the form $2^a\cdot 1033^b\cdot 119513^c$ where $0\leq a \leq 3$, $0\leq b\leq 1$, and $0\leq c \leq 1$. We see that there are $4$ choices for $a$, $2$ choices for $b$, and $2$ choices for $c$. By the Multiplication principle, the number of factors is $4\cdot 2 \cdot 2=16$.