Let n = 2³ × 3² × 5. How many factors does n have, counting 1 and n?

Every composite number is a product of primes in exactly one way, apart from the order (the Fundamental Theorem of Arithmetic). Find the primes with a factor tree or by repeated division, and write repeated primes as powers. A number ends in $0$ only if its primes include both $2$ and $5$. A number $2^a \times 3^b \times 5^c$ has $(a + 1)(b + 1)(c + 1)$ factors, since each power can be chosen from $0$ up to its top value. A perfect square has only even powers.