Question

The exponent of 12 in $100!$ is
A) 32
B) 48
C) 97
D) None of these

Answer 1

Hint:
Here, we will rewrite the given factorial in the form of the prime powers. We will then use the formula of the exponent of any prime number in factorial to find the exponent of 2 and 3 in the given factorial. We will solve it further to find the exponent in the given factorial.

Formula Used:
The exponent of any prime number $p$ in $n!$ is given by the formula $\left[ {\dfrac{n}{p}} \right] + \left[ {\dfrac{n}{{{p^2}}}} \right] + \left[ {\dfrac{n}{{{p^3}}}} \right] + ..... + \left[ {\dfrac{n}{{{p^k}}}} \right]$ where $k$ must be chosen such that ${p^{k + 1}} > n$ and $\left[. \right]$ represents the greatest integer function.

Complete step by step solution:
We are given a factorial $100!$.
We will write the given factorial in the form of prime powers as $12 = {2^2} \times 3$.
First, we will consider the factorial in terms of the power of 2.
Now using the formula $\left[ {\dfrac{n}{p}} \right] + \left[ {\dfrac{n}{{{p^2}}}} \right] + \left[ {\dfrac{n}{{{p^3}}}} \right] + ..... + \left[ {\dfrac{n}{{{p^k}}}} \right]$, we get
The exponent of 2 in $100! = \left[ {\dfrac{{100}}{2}} \right] + \left[ {\dfrac{{100}}{{{2^2}}}} \right] + \left[ {\dfrac{{100}}{{{2^3}}}} \right] + \left[ {\dfrac{{100}}{{{2^4}}}} \right] + \left[ {\dfrac{{100}}{{{2^5}}}} \right] + \left[ {\dfrac{{100}}{{{2^6}}}} \right]$
Applying the exponents on the terms of the denominator, we get
$ \Rightarrow $ The exponent of 2 in $100! = \left[ {\dfrac{{100}}{2}} \right] + \left[ {\dfrac{{100}}{4}} \right] + \left[ {\dfrac{{100}}{8}} \right] + \left[ {\dfrac{{100}}{{16}}} \right] + \left[ {\dfrac{{100}}{{32}}} \right] + \left[ {\dfrac{{100}}{{64}}} \right]$
Simplifying the fractions, we get
$ \Rightarrow $ The exponent of 2 in $100! = \left[ {50} \right] + \left[ {25} \right] + \left[ {12.5} \right] + \left[ {6.25} \right] + \left[ {3.125} \right] + \left[ {1.5625} \right]$
Since $\left[ . \right]$ represents the greatest integer function, we get
$ \Rightarrow $ The exponent of 2 in $100! = 50 + 25 + 12 + 6 + 3 + 1$
Adding the terms, we get
$ \Rightarrow $ The exponent of 2 in $100! = 97$
Since the exponent of $2$ is in the power of $2$, we get
$ \Rightarrow $ The exponent of 2 in $100! = \dfrac{{97}}{2}$
Dividing the terms, we get
$ \Rightarrow $The exponent of 2 in $100! = 48.5 \approx 48$
Now using the formula $\left[ {\dfrac{n}{p}} \right] + \left[ {\dfrac{n}{{{p^2}}}} \right] + \left[ {\dfrac{n}{{{p^3}}}} \right] + ..... + \left[ {\dfrac{n}{{{p^k}}}} \right]$, we get
$ \Rightarrow $ The exponent of 3 in $100! = \left[ {\dfrac{{100}}{3}} \right] + \left[ {\dfrac{{100}}{{{3^2}}}} \right] + \left[ {\dfrac{{100}}{{{3^3}}}} \right] + \left[ {\dfrac{{100}}{{{3^4}}}} \right]$
Applying the exponents on the terms of the denominator, we get
$ \Rightarrow $ The exponent of 3 in $100! = \left[ {\dfrac{{100}}{3}} \right] + \left[ {\dfrac{{100}}{9}} \right] + \left[ {\dfrac{{100}}{{27}}} \right] + \left[ {\dfrac{{100}}{{81}}} \right]$
Simplifying the fractions, we get
$ \Rightarrow $ The exponent of 3 in $100! = \left[ {33.33} \right] + \left[ {11.11} \right] + \left[ {3.7037} \right] + \left[ {1.2345} \right]$
Since $\left[ . \right]$ represents the greatest integer function, we get
$ \Rightarrow $ The exponent of 3 in $100! = 33 + 11 + 3 + 1$
Adding the terms, we get
$ \Rightarrow $ The exponent of 3 in $100! = 48$
We will get the exponent as 12 if the power of 2 occurs twice and the power of 3 occurs once.
Thus 48 times the exponent of 12 in the factorial $100!$.
Therefore, the exponent of 12 in $100!$ is 48.

Thus, option (B) is the correct answer.

Note:
We should note that the exponent of any prime number $p$ in $n!$ is applicable only in the case of prime numbers. If the given number is a composite number then it has to be written in the form of a prime number. A prime is a number which has only 2 factors that is the number itself and 1. The function that is rounding off the real number down to the integer less than the number is known as the greatest integer function.