Method for Counting the Divisors of a number

Question

I need to find the number of divisors of 600. Is there any other way to solve the problem, apart from writing them down and counting??

Answer 1

In general, it is very easy to write down the number of divisors if you know the prime factorization. Let's use a smaller example, say $60$.

As $60 = 2^2 \cdot 3 \cdot 5$, we can have either $0,1,$ or $2$ factors of $2$, either $0$ or $1$ factor of $3$, and either $0$ or $1$ factor of $5$. So in total, we have $3 \cdot 2 \cdot 2 = 12$ divisors of $60$. You'll note that this is of the form $(2+1)(1+1)(1+1)$, the product of one more than the exponents of the primes in the factorization.

Answer 2

As $600=2^3\cdot3^1\cdot5^2$

the number of divisors $=(3+1)(1+1)(2+1)$

See also: Divisor Function

Answer 3

$600=2^3\cdot 3\cdot 5^2$.

So any divisor of $600$ will be of the form $2^a\cdot 3^b\cdot 5^c$ where $0\le a\le 3$, $0\le b \le 1$, and $0\le c\le 2$.