Question

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

$9600 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5 \times 5$
$9600 = {2^7} \times {3^1} \times {5^2}$……………………………… (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$\left( {p + 1} \right) \times \left( {q + 1} \right) \times \left( {r + 1} \right)........... \times \left( {m + 1} \right)$.
$\Rightarrow \left( {7 + 1} \right) \times (1 \times 1) \times (2 \times 1)$
$\Rightarrow 8 \times 2 \times 3 = 48$