Answer Show 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 getThe 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.
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In this article, we will learn how to find the highest power of a number in a factorial. We will look at the three different variations of questions based on this concept that you can come across on the GMAT. So, let us get started. The first variety of question on this concept is – 1. How to find the highest power of a prime number in a factorial.Let us take an example to understand this. Say, we need to find the highest power of 3 in 20! In the exam, they can ask you this question in two ways: Question 1. AIf 20! contains 3k, where k is a positive integer, what is the highest value of k? Or they can ask the question as show below: Question 1.BWhat is the highest power of 3 in 20!? The solution is the same for either of the above questions and there are two ways to solve it. We will first solve it using method 1 which is Brute Force method, where we simply count the number of 3s. We’ll then analyse the advantages and disadvantages of this method and then move to a better method (method 2). SolutionMethod 1
Step 1
Step 2
So,
Step 3
Disadvantage of Method 1In this question, it was easy to find the highest power of 3, using method 1, because the factorial value was small. However, this method becomes tedious when the factorial numbers are high. For example, instead of 20! If we had 200! then solving the question using the above method would have taken considerable time. So, we’ll use another method to solve this type of questions and we would recommend you use the same. Method 2
Now, let us look at the second variety of questions. 2. How to find the highest power of a power of prime number in a factorialIn the last section, we learned how to find the highest power of a prime number in a factorial. In this section, we will extend the same concept to find the highest power of a power of prime number( i.e. a number in the form of , where p is a prime number and q is a positive integer greater than 1) in a factorial. Let us understand with an example. Question 2What is the highest power of 8 in 70! ? Common Mistake:On seeing this question, a lot of students follow the approach shown below:
However, this is INCORRECT, because 8 is not a prime number and we cannot directly divide by a non-prime number to find the highest power of it in a factorial. The following section explains the correct step-by-step procedure of solving such type of questions. SolutionStep 1
Step 2
Step 3
So, one major learning here is that don’t divide the factorial by a non-prime number. First break the number down into its prime factorized form and then find the highest power of that number. And after that figure out what will be the highest power of that non-prime number. Till now, we have seen how to find the highest power of a prime number or a power of a prime number, in the next section, we will see how to find the highest power of a number that has two distinct prime number. Recommended: 3 Important Properties of Prime Numbers 3. How to find the highest power of a number that has two distinct prime numbersFinding the highest power in this case is only a little bit different from the last section as instead of one prime factor there will be more than one prime factor i.e. the number will be in the form of , where and are prime numbers. So, with the help of an example, let us understand how to solve this type of question. Question 3What is the highest power of 10 in 100!? SolutionStep 1
So, now we know that to make one 10 we need one 2 and one 5. So, in our last step, let’s see how many 10s we can make in 100! Step 2
Takeaway
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