Finding out the number of divisors

Question

The number of divisors of 9600 including 1 and 9600 are??

Is there any other way of doing this rather than manually finding the divisors?

Answer 1

Find the prime factorisation of 9600 first. At a glance, $9600=2^7*3*5^2$ So the total number of divisors= $8*2*3=48$.

Explanation: after finding the prime factorisation, add 1 to each exponent and multiply to obtain the new number, which is the number of divisors. This works because any divisor of 9600 can be expressed in the form $2^a*3^b*5^c$. And, if we just consider a, a can take on values from 0 to 7, so it can take on 8 possible values. Similarly, b takes on values from 0 to 1, and c from 0 to 2.

From which we arrive at 8*2*3=48, using multiplication principle to deduce the total number of divisors.

Answer 2

Let $n\in\mathbb{N}$. Then by Fundamental Theorem of Arithmetic we can write $n\in \mathbb{N}, n\neq 1$ by $n=p_1^{a_1}p_2^{a_2}\dots p_k^{a_k}$ where $p_1,p_2,\dots p_k$ are prime and $a_k\in\mathbb{N}$, $k=1,2,...,k$. After that use the formula for number of divisor $(a_1+1)(a_2+1)\cdots (a_k+1)$. (It is easy to check the formula).