Question

The number of positive divisors of 768 is
A.17
B.18
C.19
D.20

Answer 1

Hint: Here, we have to find the number of positive divisors. We will find the divisors of the number using the formula. Then by using the formula for the number of divisors we will get the required number.

Formula Used:
We will use the following formulas:
1.Number $ = {a^p}{b^q}.......$, where $a,b,.....$ are prime factors.
2.Number of divisors $ = (p + 1)(q + 1)$

Complete step-by-step answer:
We have to find the number of positive divisors.
First we will write the number in terms of prime factors.
So, now we have
$ 2\left| \!{\underline {\,
  {768} \,}} \right. \\
  2\left| \!{\underline {\,
  {384} \,}} \right. \\
  2\left| \!{\underline {\,
  {192} \,}} \right. \\
  2\left| \!{\underline {\,
  {96} \,}} \right. \\
  2\left| \!{\underline {\,
  {48} \,}} \right. \\
  2\left| \!{\underline {\,
  {24} \,}} \right. \\
  2\left| \!{\underline {\,
  {12} \,}} \right. \\
  2\left| \!{\underline {\,
  6 \,}} \right. \\
  3\left| \!{\underline {\,
  3 \,}} \right. \\
  1\left| \!{\underline {\,
  1 \,}} \right. \\ $
Therefore, we can write
$768 = {2^8} \times 3$
The positive divisor of $768$can be any of the term of the expansion \[({2^0} + {2^1} + {2^2} + {2^3} + {2^4} + {2^5} + {2^6} + {2^7} + {2^8})({3^0} + {3^1})\].
Now we will find the total number of divisors .
Number of divisors $ = (p + 1)(q + 1)$
$ \Rightarrow $Number of divisors \[ = \left( {8 + 1} \right)\left( {1 + 1} \right)\]
Adding the terms, we get
$ \Rightarrow $Number of divisors \[ = \left( 9 \right)\left( 2 \right)\]
Multiplying the terms, we get
$ \Rightarrow $Number of divisors $ = 18$
Therefore, the numbers of positive divisors are 18.

Note: An integer $d$ is called a divisor of an integer $n$ , if $d$ evenly divides $n$ without leaving a remainder, that is if there is an integer $k$ such that $n = dk$. Divisor of $n$ is also called a factor of $n$ . A positive divisor of $n$ which is different from $n$ is called a proper divisor or an aliquot part of $n$ . If a number does not evenly divide n but leaves a remainder it is called an aliquant part of $n$ . If $d$ is a divisor of $n$we also say that $n$ is divisible by $d$ , or that $n$ is a multiple of $d$.