Question

The exponent of $3$ in $100!$ is
A.47
B.48
C.49
D.46

Answer 1

Hint: By the exponent of a number in a factorial of another number, we mean the degree of the number by which the factorial can be divided to give natural numbers.

Complete step by step Answer:
Here, we have to find the maximum degree of $3$that can divide$100$. We will also use the Greatest Integer function to find the exponent.

We know that $n! = n \times (n - 1) \times (n - 2) \times (n - 3).... \times 3 \times 2 \times 1$

So $100!$can be written as
$100! = 100 \times 99 \times 98 \times 97.... \times 3 \times 2 \times 1$

We know that exponent of a prime number $p$in $n!$is
$\left[ {\dfrac{n}{p}} \right] + \left[ {\dfrac{n}{{{p^2}}}} \right] + \left[ {\dfrac{n}{{{p^3}}}} \right] + \left[ {\dfrac{n}{{{p^4}}}} \right]$….. and so on.

Where $[x]$is the greatest integer less than or equal to $x$.

Therefore, the exponent of $3$is

$$\left[ {\dfrac{{100}}{{{3^1}}}} \right] + \left[ {\dfrac{{100}}{{{3^2}}}} \right] + \left[ {\dfrac{{100}}{{{3^3}}}} \right] + \left[ {\dfrac{{100}}{{{3^4}}}} \right] \\\ = \left[ {\dfrac{{100}}{3}} \right] + \left[ {\dfrac{{100}}{9}} \right] + \left[ {\dfrac{{100}}{{27}}} \right] + \left[ {\dfrac{{100}}{{81}}} \right] \\\ = [33.33] + [11.11] + [3.703] + [1.234] \\\ = 33 + 11 + 3 + 1 \\\ = 48 \\\$$

We only consider the degree to be $4$because beyond that the value of the exponent reaches $0$, for example: $\left[ {\dfrac{{100}}{{{3^5}}}} \right] = \left[ {\dfrac{{100}}{{243}}} \right] = [0.411] = 0$. Hence, the maximum degree is $4$.

The greatest integer function rounds down the real number to its nearest integer less than or equal to the real number.

Hence, the exponent of $3$in $100!$is $48$.

Thus, the answer is option B.

Note: We need to remember that an exponent can be calculated for only prime numbers and we have to use the greatest integer function to solve the problem. If we are given a non-prime number, we will have to prime factorize the number to find the prime numbers and then find the exponent.
Also, we can look at the question in another way.
We can find the number of multiples of different degrees of the prime number in between $1\& n$ and add all of them to find the exponent.

In this question, there are $33$multiples of 3 in between $1\& 100$, 11 multiples of 9, 3 multiples of 27 and 1 multiple of 81. Hence, $33 + 11 + 3 + 1 = 48$