Question

Find the number of divisors of 3600 excluding 1 and itself-
A. 44
B. 45
C. 47
D. 43

Answer 1

Hint: The formula for calculating the number of factors will be used in this question. The number of divisors for a given number of the form ${2^{\text{a}}} \times {3^{\text{b}}} \times {5^{\text{c}}}...$ is given by-
${\text{D}}\left( {\text{n}} \right) = \left( {{\text{a}} + 1} \right)\left( {{\text{b}} + 1} \right)\left( {{\text{c}} + 1} \right)...$

Complete step-by-step answer:
We have to find the number of divisors of 3600 excluding 1 and itself, so we will first do the prime factorization of 3600 as follows-
$\begin{align}
  &2\left| {\underline {3600} } \right. \\
  &2\left| {\underline {1800} } \right. \\
  &\left. 2 \right|\underline {900} \\
  &2\left| {\underline {450} } \right. \\
  &3\left| {\underline {225} } \right. \\
  &3\left| {\underline {75} } \right. \\
  &5\left| {\underline {25} } \right. \\
  &5\left| {\underline 5 } \right. \\
  &\;\;\left| 1 \right. \\
\end{align} $

So, we can write that-
$\begin{align}
  &3600 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5 \\
  &3600 = {2^4} \times {3^2} \times {5^2} \\
\end{align} $

Using the formula for the number of divisors, we get that-
$D(n) = (4 + 1)(2 + 1)(2 + 1)$
$D(n) = (5)(3)(3) = 45$
But these are the total number of divisors. The number of divisors of 3600 excluding 1 and itself are $(45 - 2) = 43$

This is the required answer, the correct option is D. 43

Note: The most common mistake is that students find the total number of divisors and mark them as the answer, which is wrong. We need to read what has been asked carefully and then mark the answer. Also, students need to remember the formula for finding the number of divisors.