Question

Find the number of divisors of 9600 including 1 and 9600.$\begin{gathered} \left( a \right){\text{ 60}} \\ \left( b \right){\text{ 58}} \\ \left( c \right){\text{ 48}} \\ \left( d \right){\text{ 46}} \\ \end{gathered}$

Hint- Find all the factors of the given number and then try to evaluate the total number of divisors from these factor representations.
First find all the factors of 9600 are
$9600 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5 \times 5$
We can also write this down as
$9600 = {2^7} \times {3^1} \times {5^2}$â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (1)
Now if a number has factors of the form ${a^p} \times {b^q} \times {c^r}.........{z^m}$then the divisors of that numbers can be written as$\left( {p + 1} \right) \times \left( {q + 1} \right) \times \left( {r + 1} \right)........... \times \left( {m + 1} \right)$.
Using the same concept in equation (1) we get the total number of divisors of 9600 and it includes 1 and the number itself.
$\Rightarrow \left( {7 + 1} \right) \times (1 \times 1) \times (2 \times 1)$
$\Rightarrow 8 \times 2 \times 3 = 48$
Hence the correct option is (c).
Note- Whenever we face such a problem statement the key concept involved is simply to find all the factors of the given number now through those factors we can directly find the total number of divisors using the above mentioned concept. The note point here is this number of divisors obtained will always include 1 and the number itself.