Divisors of 7200

Text

How many positive divisors does the number $7200$ have?

Solution

We factor into primes: $7200 = 72 \cdot 100 = (2^{3}\cdot 3^{2}) \cdot (2^{2}\cdot 5^{2}) = 2^{5}\cdot 3^{2}\cdot 5^{2}.$ Every divisor has the form $2^{a}\,3^{b}\,5^{c}$ with $a\in\{0,\dots,5\}$ ($6$ choices), $b\in\{0,1,2\}$ ($3$ choices), $c\in\{0,1,2\}$ ($3$ choices). By the product rule: $N = (5+1)(2+1)(2+1) = 6\cdot 3\cdot 3 = \boxed{54}.$