Question

The number of non-trivial divisors of 2160 is
A.40
B.39
C.38
D.18

Answer 1

Hint: We will use the concept of prime factorization to find the number of divisors of 2160. We will subtract 2 from the total number of divisors to find the number of non-trivial divisors of 2160.

Formulas used: If a number \[n\] has the prime factorization as \[n = {p^a} \times {q^b} \times {r^c}\] . Then the number of divisors of \[n\] is \[\left( {a + 1} \right)\left( {b + 1} \right)\left( {c + 1} \right)\].

Complete step-by-step answer:
We know that a divisor of a number \[n\] is any number that leaves no remainder when \[n\] is divided by it (or completely divides \[n\]).
We will find the total number of divisors of 2160. For this, we will first find the prime factorization of 2160:

We can see that \[2160 = {2^4} \times {3^3} \times {5^1}\].
We will calculate the total number of divisors of 2160. We will substitute 4 for \[a\], 3 for \[b\] and 1 for \[c\] in the formula for total number of divisors:
\[\left( {4 + 1} \right)\left( {3 + 1} \right)\left( {1 + 1} \right) = 5 \cdot 4 \cdot 2 = 40\]
The total number of divisors of 2160 is 40.
We know that the trivial divisors of a number are 1 and the number itself. All divisors of a number other than the trivial divisors are known as the non-trivial divisors of that number. So the number of non-trivial divisors will be 2 less than the total divisors of a number. We will find the non-trivial divisors of 2160:
\[40 - 2 = 38\]
We have calculated that the total number of non-trivial divisors of 2160 is 38.
$\therefore $ Option C is the correct option.
Note: We can also count the non-trivial divisors by listing out all the divisors of 2160 and counting them:
\[\begin{array}{*{20}{c}}\begin{array}{l}1 \times 2160\\2 \times 1080\\3 \times 720\\4 \times 540\\5 \times 432\end{array}&\begin{array}{l}6 \times 360\\8 \times 270\\9 \times 240\\10 \times 216\\12 \times 180\end{array}&\begin{array}{l}15 \times 144\\16 \times 135\\18 \times 120\\20 \times 108\\24 \times 90\end{array}&\begin{array}{l}27 \times 80\\30 \times 72\\36 \times 60\\40 \times 54\\45 \times 48\end{array}\end{array}\]
We can see from the above list that 2160 has 38 non-trivial divisors.