How to Find Number of Trailing Zeros in a Factorial or Product

Wednesday, March 14th, 2018


How to Find Number of Trailing Zeros in a Factorial or Product

In this Number of trailing Zeros blog post, I would like to cover these two ideas:

  • Number of trailing zeroes in a Product or Expression

  • Number of trailing zeroes in a factorial (n!)

But before I begin, let us first try to understand what exactly are ‘trailing zeroes’. It is nothing else but the number of zeroes at the end. I do not want to sound pedantic but on many occasions when you see a question which asks about, “What is the number of zeroes in ___” it is incorrect, because it should actually say – “What is the number of trailing zeroes?” or “What is the number of ending zeroes?” It is virtually impossible to predict the exact number of zeroes without actually doing the calculation and finding out the answer.

Just to clarify, 170130000 has 5 zeroes but 4 trailing / ending zeroes.

In questions based on these ideas, you should assume that the examiner is asking about trailing zeroes unless specified otherwise.

Number of trailing zeroes in a Product or Expression

If we look at a number N, such that

TRAIL-1

Number of trailing zeroes is the Power of 10 in the expression or in other words, the number of times N is divisible by 10.

For a number to be divisible by 10, it should be divisible by 2 & 5.

Consider a number N, such that

TRAIL-2 

For the number to have a zero at the end, both a & b should be at least 1.

If you want to figure out the exact number of zeroes, you would have to check how many times the number N is divisible by 10.

When I am dividing N by 10, it will be limited by the powers of 2 or 5, whichever is lesser.

Number of trailing zeroes is going to be the power of 2 or 5, whichever is lesser.

Let us look at an example to further illustrate this idea.

Question 1: What is number of trailing zeroes in 12000?

12000 = 25 x 3 x 53

When I divide it by 10, it would be divisible exactly thrice because I have only three 5s.

In this case, number of 5s has become the limiting factor and so, the power of 5, which is 3 is the answer.

Tip: The power of 5 will be the limiting factor in most cases of continuous distribution. It will happen because 5 is less likely to occur than 2.

 

Question 2: Find out the number of zeroes at the end of N

TRAIL-3
 
Looking at the expression, we can say that the power of 5 will be the limiting factor.

All we need to do is to figure out the number of 5s in the expression.

11, 22, 33, 1717, 8989,… will not give us any 5s.

55 will give us five 5s.

1010 will give us ten 5s.

1515 will give us fifteen 5s.

And so on.

So, the total number of 5s that I have is

TRAIL-4
 
But I have made a mistake in the above calculation.

I have assumed that 2525 will give me twenty-five 5s but that is incorrect.

It is incorrect because 2525 = 550 and will actually give me 50 5s.

Other errors are

5050 will actually give me 100 5s, whereas I have considered only 50 5s.

7575 will actually give me 150 5s, whereas I have considered only 75 5s.

100100 will actually give me 200 5s, whereas I have considered only 100 5s.

Considering the above, I have made an error of = 25 + 50 + 75 + 100 = 250 5s.

So the total number of 5s that I have are 1050 + 250 = 1300.

So the number of trailing zeroes at the end of the expression is 1300

Number of trailing zeroes in a factorial (n!)

Number of trailing zeroes in n! = Number of times n! is divisible by 10 = Highest power of 10 which divides n! = Highest power of 5 in n!

The question can be put in any of the above ways but it can be answered using the simple formula given below:

TRAIL-5
 
{ [x] is the greatest integer function. [4.99] = 4, [4.01] = 4, [- 4.99] = 5, [-4.01] = 5}

 The above formula gives us the exact number of 5s in n! because it will take care of all multiples of 5 which are less than n. Not only that it will take care of all multiples of 25, 125, etc. (higher powers of 5).

Tip: Instead of dividing by 25, 125, etc. (higher powers of 5); it would be much faster if you divided by 5 recursively.

Let us use this to solve a few examples:

Question 3: What is the number of trailing zeroes in 23! ?

[23/5 ]= 4. It is less than 5, so we stop here.

The answer is 4.

 

Question 4: What is the number of trailing zeroes in 123! ?

[123/5] = 24

Now we can either divided 123 by 25 or the result in the above step i.e. 24 by 5.

[24/5 ]= 4. It is less than 5, so we stop here.

The answer is = 24 + 4 = 28

 

Question 5: What is the number of trailing zeroes in 1123!?

[1123/5] = 224

[224/5] = 44

[44/5] = 8

[8/5]=1. It is less than 5, so we stop here.

The answer is = 224 + 44 + 8 + 1 =277

 

Finding out ‘n’ when number of trailing zeros is known

Question 6: Number of trailing zeroes in n! is 13. n =?

There is no standard formula for such type of questions but they can be solved by a little bit of hit and trial.

I need to get 13 trailing zeroes which I will definitely get from 65!

But it will have some extra zeroes in the end because of higher powers of 5.

So, I will consider the previous multiple of 5, which in this case is 60.

Trailing zeroes in 60! = [60/5] + [60/25] = 12 + 2 = 14

I got 14 but I want to get 13, so I will consider the previous multiple of 5, which in this case is 55.

Trailing zeroes in 55! = [55/5] + [55/25] = 11 + 2 = 13

So, the valid values of n! are 55!, 56!, 57!, 58!, 59!

 

Question 7: Number of zeroes in n! is 23. n =?

Trailing zeroes in 100! = [100/5] + [100/25 ] = 20 + 4 = 24 {Too high. Consider previous multiple}

Trailing zeroes in 95! = [95/5] + [95/25] = 19 + 3 = 22 {Too low. Consider next multiple}

As you can see from above, we would end up in a loop.

This will happen because there is no valid value of n for which n! will have 23 zeroes in the end.

You can also see Learning New Words – Why, How and Strategies

Writing the end of the posts is the hardest part for me because I cannot figure out a decent way to end it. It seems like I am in flow when I am writing this and it gets broken at the end. Read the wiki page, you will learn something much more important than Math.

Other posts related to Quantitative Aptitude – Number Systems

Divisibility Rules for CAT Quantitative Aptitude Preparation
Remainders (Quantitative Aptitude) for CAT Exam Preparation – Free PDF for Download
Baisc Idea of Remainders
Factor Theory
Determining the second last digit and the last two digits
Dealing With Factorials
Application of LCM (Lowest Common Multiple) in solving Quantitative Aptitude Problems

CAT Questions related to Quantitative Aptitude – Number Systems

All questions from CAT 2017 Quantitative Aptitude – Number Systems
Quantitative Aptitude – Number Systems – Q1: If the product of three consecutive positive integers is 15600 then the sum of the squares of these integers is
Quantitative Aptitude – Number Systems – Q2: If a, b, c are three positive integers such that a and b are in the ratio 3 : 4 while b and c are in the ratio 2:1, then which one of the following is a possible value of (a + b + c)?
Quantitative Aptitude – Number Systems – Q3: The numbers 1, 2,…,9 are arranged in a 3 X 3 square grid in such a way that each number occurs once and the entries along each column, each row, and each of the two diagonals add up to the same value.

Online Coaching Course for CAT 2018

a) 750+ Videos covering entire CAT syllabus
b) 2 Live Classes (online) every week for doubt clarification
c) Study Material & PDFs for practice and understanding
d) 10 Mock Tests in the latest pattern
e) Previous Year Questions solved on video

Know More about Online CAT Course    

How to Find Number of Trailing Zeros in a Factorial or Product

4.9 (98.7%) 108 votes



If you Like this post then share it!


2 responses to “How to Find Number of Trailing Zeros in a Factorial or Product”

  1. […] Determining the second last digit and the last two digits Baisc Idea of Remainders Factor Theory How to Find Number of Trailing Zeros in a Factorial or Product Dealing With […]