Question

find the highest power of 10 in 200 factorial​

Answer 1

Answer:

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Highest power of a number in Factorial | GMAT Quant Prep

July 27, 2020

Calculate the highest power of a number in a Factorial?

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. A

If 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.B

What 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).

Solution

Method 1

We need to find the highest power of 3 in 20!

Step 1

Firstly, we will jot down all the multiples of 3 which are less or equal to 20.

Multiples of 3 which are less than or equal to 20 are 3, 6, 9, 12, 15, and 18.

Step 2

We will prime factorize the multiples of 3 to get the greatest power of 3 in each of them.

So,

3 = 31

6 = 21*31

9 = 32

12 = 22*31

15 = 31*51

18 = 21*32

Step 3

We will add up all the highest powers of 3 obtained from each of its multiple.

So, the highest power of 3 in 20! = 1 + 1+ 2+ 1+ 1+ 2 = 8

And thus, k = 8

Answer 2

Concept:

Factorial of a number n is given as

n!=n×(n-1)×(n-2).....2×1

For calculating the highest power of 'p' in a factorial 'n'

$[{\frac{n}{p}]+[{\frac{n}{p^{2} }} ]+[\frac{n}{p^{3} } ]+....$

Given:

Highest power of 10 in 200!

To find:

The highest power of 10 in 200!

Solution:

We know that

$[{\frac{n}{p}]+[{\frac{n}{p^{2} }} ]+[\frac{n}{p^{3} } ]+....$

We need to find the highest power of 10 in 200 factorial

Therefore, p=10 and n=200

$[\frac{200}{10} ]+[\frac{200}{10^{2} }]+ [\frac{200}{10^{3} }]+[\frac{200}{10^{4} } ].....$

⇒$[\frac{200}{10} ]+[\frac{200}{100} }]+ [\frac{200}{1000 }]+[\frac{200}{10000} } ].....$

⇒$20+2+0$             [the values after 2nd value the values are approx 0]

⇒$22$

Therefore the highest power of 10 in 200! is 22.