Question

The number of non-trivial divisors of \[{\text{2160}}\] is
A. \[{\text{40}}\]
B. \[39\]
C. \[38\]
D. \[18\]

Answer 1

Hint: First of all calculate the total number of divisors of \[2160\], by using the concept that let \[{\text{p}}\]be any number with factor \[{{\text{x}}^{\text{r}}}{{\text{y}}^{\text{s}}}\] then its total factor will be \[{\text{(r + 1)(s + 1)}}\]. Non – trivial factors are the all the factors of the number except \[1\] and number itself. Because \[1\] is trivial factor and number itself is it’s improper factor. So, in order to can calculate non-trivial factors we just need to calculate [all factors \[{\text{ - 2}}\]]

Complete step by step answer:

First of all calculating all the factors of \[{\text{2160}}\],
In the factorization we calculate all the possible numbers through which the given number is divisible and after finding all it’s possible factors we write it in a representation of all the prime possible factors such as \[{{\text{x}}^{\text{r}}}{{\text{y}}^{\text{s}}}\]
On factorising, we get,
\[
  \begin{array}{*{20}{c}}
  {\text{3}}&{{\text{2160}}}
\end{array} \\
  \begin{array}{*{20}{c}}
  {\text{3}}&{{\text{720}}}
\end{array} \\
  \begin{array}{*{20}{c}}
  {\text{3}}&{{\text{240}}}
\end{array} \\
  \begin{array}{*{20}{c}}
  {\text{5}}&{{\text{80}}}
\end{array} \\
  \begin{array}{*{20}{c}}
  {\text{2}}&{{\text{16}}}
\end{array} \\
  \begin{array}{*{20}{c}}
  {\text{2}}&{\text{8}}
\end{array} \\
  \begin{array}{*{20}{c}}
  {\text{2}}&{\text{4}}
\end{array} \\
  \begin{array}{*{20}{c}}
  {\text{2}}&{\text{2}}
\end{array} \\
  {\text{1}} \\
 \]
So, \[{\text{2160}}\] = \[{{\text{2}}^{\text{4}}}{\text{.}}{{\text{3}}^{\text{3}}}{\text{.}}{{\text{5}}^{\text{1}}}\]
After factorizing the given number use the concept mentioned in the hint about how to calculate non-trivial factors.
So first calculate total factors and then subtract the number itself and one from total factors as it will be our answer.
Hence, there total number of divisors are \[{\text{(4 + 1)(3 + 1)(1 + 1) = 40}}\]
Now , to calculate non trivial factors just proceed with [all factors \[{\text{ - 2}}\]]
So, it will be
\[ \Rightarrow 40 - 2 = 38\]
Hence, option ( c) is our required correct answer.

Note: Every integer and its negation is a divisor of itself. Integers divisible by 2 are called even, and integers not divisible by 2 are called odd. 1, −1, n, and −n are known as the trivial divisors of n. A divisor of n other than a trivial divisor is known as a non-trivial divisor or a strict divisor.