Question

The total number of proper divisors of 38808 excluding 1 and itself is
(a) 72
(b) 70
(c) 69
(d) 71

Answer 1

Hint: We solve this problem by using the prime factorisation method. The prime factorisation method is nothing but expressing the given numbers in terms of product of prime numbers. Then we use the standard formula for finding the number of divisors that is if a number is of the form
\[n={{a}_{1}}^{{{x}_{1}}}\times {{a}_{2}}^{{{x}_{2}}}\times {{a}_{3}}^{{{x}_{3}}}\times .........\times {{a}_{n}}^{{{x}_{n}}}\]
Where \[{{a}_{1}},{{a}_{2}},......{{a}_{n}}\] are prime numbers
Then the formula for number of divisors is given as
\[\text{number of divisors}=\left( {{x}_{1}}+1 \right)\left( {{x}_{2}}+1 \right)\left( {{x}_{3}}+1 \right).....\left( {{x}_{n}}+1 \right)\]

Complete step by step answer:
We are given that number as 38808.
Now, let us use the prime factorisation method.
We know that the prime factorisation method is nothing but expressing the given numbers in terms of product of prime numbers.
Now, by dividing the given number with first prime number 2 we get
\[\Rightarrow 38808=2\times 19404\]
Here, we can see that the number can be again divided by 2then we get
\[\begin{align}
  & \Rightarrow 38808={{2}^{2}}\times 9702 \\
 & \Rightarrow 38808={{2}^{3}}\times 4851 \\
\end{align}\]
Here we can see that the number 4851 can no longer be divided by number 2.
So, let us go for next prime number that 3
By dividing the number 4851 by 3 we get
\[\begin{align}
  & \Rightarrow 38808={{2}^{3}}\times 3\times 1617 \\
 & \Rightarrow 38808={{2}^{3}}\times {{3}^{2}}\times 539 \\
\end{align}\]
Now, we can see that the number 593 can no longer be divided by 3
So, let us go for next prime number that is 5 but, the number 539 cannot be divisible by 5.
Now, let us go for next prime number that is 7
Now, by dividing the number 539 with 7 then we get
\[\begin{align}
  & \Rightarrow 38808={{2}^{3}}\times {{3}^{2}}\times 7\times 77 \\
 & \Rightarrow 38808={{2}^{3}}\times {{3}^{2}}\times {{7}^{2}}\times 11 \\
\end{align}\]
Here, we can see that the number is completely represented as the product of prime numbers.
Let us assume that the number of divisors of 38808 as \['k'\]
We know that if a number is of the form
\[n={{a}_{1}}^{{{x}_{1}}}\times {{a}_{2}}^{{{x}_{2}}}\times {{a}_{3}}^{{{x}_{3}}}\times .........\times {{a}_{n}}^{{{x}_{n}}}\]
Where \[{{a}_{1}},{{a}_{2}},......{{a}_{n}}\] are prime numbers
Then the formula for number of divisors is given as
\[\text{number of divisors}=\left( {{x}_{1}}+1 \right)\left( {{x}_{2}}+1 \right)\left( {{x}_{3}}+1 \right).....\left( {{x}_{n}}+1 \right)\]
Now, by using the above formula to given number 38808 we get
\[\begin{align}
  & \Rightarrow k=\left( 3+1 \right)\left( 2+1 \right)\left( 2+1 \right)\left( 1+1 \right) \\
 & \Rightarrow k=4\times 3\times 3\times 2=72 \\
\end{align}\]
So, the number has 72 divisors.
But, we are asked to find the number of divisors after excluding 1 and itself.
Let us assume that the required number of divisors as \['p'\]
So, we get the required number of divisors by subtracting 2 from total divisors that is
\[\Rightarrow p=72-2=70\]
Therefore the required number of divisors is 70.

So, the correct answer is “Option b”.

Note: Students may make mistakes in finding the required divisors.
We have the total number of divisors as
\[\Rightarrow k=72\]
Here, of those 72 divisors there are 1 and 38808 also included.
But we need to exclude 1 and 38808 which we get by subtracting 2 from total divisors which gives the required divisors as 70.