Question

Find the number of divisors of $10800$.
A. 57
B. 60
C. 72
D. None of these

Answer 1

Hint: In this question we will first find all the prime factors of $n$ and write it as ${a^p},{b^q},{c^r}$ where $a,b,c$ are prime numbers i.e. $n = {a^p},{b^q},{c^r}$. Then, the number of divisors of $n$ are $\left( {p + 1} \right) \times \left( {q + 1} \right) \times \left( {r + 1} \right)$.

Complete step-by-step answer:
According to the question , the given number is $10800$ of which we have to find the number of divisors.
Hence,
$
  10800 = 3 \times 3 \times 3 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5 \\
  10800 = {3^3} \times {2^4} \times {5^2} \\
  n = {a^p},{b^q},{c^r} \\
 $
Then the number of divisors $n = \left( {p + 1} \right) \times \left( {q + 1} \right) \times \left( {r + 1} \right)$
Here ,
$
  n = 10800\;,\;a = 3\;,\;b = 2\;,\,c = 5\;,\;p = 3\;,\;q = 4\;,\;c = 2 \\
   \Rightarrow n = \left( {3 + 1} \right) \times \left( {4 + 1} \right) \times \left( {2 + 1} \right) \\
   \Rightarrow n = 4 \times 5 \times 3 = 60 \\
$

Note: It is advisable to remember basic formulas like the total numbers of divisors i.e. $n = \left( {p + 1} \right) \times \left( {q + 1} \right) \times \left( {r + 1} \right)$which involving into divisor of number questions as it helps in solving questions easily.