Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized for You

we will pick new questions that match your level based on your Timer History

Track Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

Re: If 73! has 16 zeroes at the end, how many zeroes will 80! [#permalink]

Show Tags

15 Feb 2013, 12:34

4

This post received KUDOS

1

This post was BOOKMARKED

2 x 5 gives you a zero at the end of the number. This means you are only looking for additional 5s (there are plenty of factors of 2 in 73! 75 has 2 5s and 80 has 1. So, there should be 3 more (this means that the answer is actually 19.

What is the source of your question?
_________________

Want to Ace the GMAT with 1 button? Start Here: GMAT Answers is an adaptive learning platform that will help you understand exactly what you need to do to get the score that you want.

If 73! has 16 zeroes at the end, how many zeroes will 80! have at the end?

A. 16 B. 17 C. 18 D. 19 E. 20

We can solve the question using info given in the stem (about 73!) but I find it easier to use direct approach of trailing zeros.

Trailing zeros: Trailing zeros are a sequence of 0's in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow.

125000 has 3 trailing zeros;

The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer n, can be determined with this formula:

\(\frac{n}{5}+\frac{n}{5^2}+\frac{n}{5^3}+...+\frac{n}{5^k}\), where k must be chosen such that 5^(k+1)>n

It's more simple if you look at an example:

How many zeros are in the end (after which no other digits follow) of 32!? \(\frac{32}{5}+\frac{32}{5^2}=6+1=7\) (denominator must be less than 32, \(5^2=25\) is less)

So there are 7 zeros in the end of 32!

The formula actually counts the number of factors 5 in n!, but since there are at least as many factors 2, this is equivalent to the number of factors 10, each of which gives one more trailing zero.

BACK TO THE ORIGINAL QUESTION:

According to above 80! has \(\frac{80}{5}+\frac{80}{25}=16+3=19\) trailing zeros (take only the quotient into account).

Re: If 73! has 16 zeroes at the end, how many zeroes will 80! [#permalink]

Show Tags

06 May 2013, 19:05

1

This post received KUDOS

A tougher question wouldn't have the comment 73! so I chose to handle this one head on, that is, looking for the number of 0's that can be constructed from 80!

Prime Factoring: you need a 2 and a 5 to make a 10 (a "zero"), and there are TONS of 2's so let's skip these and focus on the (rarer) 5s:

80! = 1*2*3*4*5*6...*78*79*80

Since there are 80 consecutive numbers, there are 16 multiples of 5 in there, but if we're prime factoring, we need to remember that SOME multiples of 5 actually contain more than just one 5. Which? 25 comes to mind -- it's got two of them! So all the multiples of 25 actually contain two 5's (ie: 50 and 75)

So, to recap, we have 16 of them, plus 3 more (the additional 5's in 25, 50, and 75), so that makes 19, and since we have more than enough 2's, we know our number will have exactly 19 zeros at the end.

Re: If 73! has 16 zeroes at the end, how many zeroes will 80! [#permalink]

Show Tags

23 Jan 2015, 21:42

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

Re: If 73! has 16 zeroes at the end, how many zeroes will 80! [#permalink]

Show Tags

24 May 2016, 22:48

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

Re: If 73! has 16 zeroes at the end, how many zeroes will 80! [#permalink]

Show Tags

27 Aug 2017, 18:57

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

Re: If 73! has 16 zeroes at the end, how many zeroes will 80! [#permalink]

Show Tags

30 Aug 2017, 22:04

2.- If 83! has 16 zeroes at the end, how many zeroes will 90! have at the end?

A. 16 B. 17 C. 18 D. 19 E. 20
_________________

claudio hurtado maturana Private lessons GMAT QUANT GRE QUANT SAT QUANT Classes group of 6 students GMAT QUANT GRE QUANT SAT QUANT Distance learning courses GMAT QUANT GRE QUANT SAT QUANT

Website http://www.gmatchile.cl Whatsapp +56999410328 Email clasesgmatchile@gmail.com Skype: clasesgmatchile@gmail.com Address Avenida Hernando de Aguirre 128 Of 904, Tobalaba Metro Station, Santiago Chile.

Re: If 73! has 16 zeroes at the end, how many zeroes will 80! [#permalink]

Show Tags

30 Aug 2017, 22:08

gmatchile wrote:

2.- If 83! has 16 zeroes at the end, how many zeroes will 90! have at the end?

A. 16 B. 17 C. 18 D. 19 E. 20

83! = 16 zeroes at the end 90! = 83! 84*85*86*87*88*89*90=(16 ceros)*(2*42)*(5*17)*(2*43)*(3*29)*(2*44)*89*(9*10)= =(10^16)*(2*5)*42*17*2*43*3*29*2*44*89*9*10=(10^16)*10*10*42*17*43*3*29*2*44*89*9=(10^18)*…… 90! Increment two zeroes at the end. 16 + 2 = 18 zeroes at the end. c)
_________________

claudio hurtado maturana Private lessons GMAT QUANT GRE QUANT SAT QUANT Classes group of 6 students GMAT QUANT GRE QUANT SAT QUANT Distance learning courses GMAT QUANT GRE QUANT SAT QUANT

Website http://www.gmatchile.cl Whatsapp +56999410328 Email clasesgmatchile@gmail.com Skype: clasesgmatchile@gmail.com Address Avenida Hernando de Aguirre 128 Of 904, Tobalaba Metro Station, Santiago Chile.

If 73! has 16 zeroes at the end, how many zeroes will 80! have at the end?

A. 16 B. 17 C. 18 D. 19 E. 20

Since we are given that 73! has 16 zeros at the end, determining the number of zeros 80! will have at the end is the same as determining the number of extra zeros that will be produced from the following product:

74 x 75 x 76 x 77 x 78 x 79 x 80

To determine the number of zeros, we need to determine the number of pairs of fives and twos that are created from the above product. Note that we need five-and-two pairs because 5 x 2 = 10, and each 10 creates an additional trailing zero.

Since we know there are going to be fewer fives than twos, let’s determine the number of fives.

80 = 2^4 x 5

75 = 5^2 x 3

We see that since there are 3 fives, there will be 3 five-two pairs, and thus 3 trailing zeros. Therefore, there will be a total of 19 trailing zeros.

Answer: D
_________________

Jeffery Miller Head of GMAT Instruction

GMAT Quant Self-Study Course 500+ lessons 3000+ practice problems 800+ HD solutions

Re: If 73! has 16 zeroes at the end, how many zeroes will 80! [#permalink]

Show Tags

02 Sep 2017, 11:23

To figure out how many trailing zeros are in 80!, one will need to be aware that the only way to get a zero at the end of a number is if that number has a 10 as a factor, and by having 10 as a factor, that number will also have 2 and 5 as factors. Effectively, all you have to do is figure out how many factors of 5 are there in that number.

Since the question is asking for 1*2*3*...*80, we know that there are 16 multiples of 5 (for example: one for 5, another one for 10, another one for 15, and so on). We are not done yet because we have to account for any factors with extra 5's. In this case we have 25, 50, and 75.

25: 5*5 -> an extra 5 50: 5*5*2 -> an extra 5 75: 5*5*3 -> an extra 5

All together we have 16 multiples of 5 plus three extra 5's for each multiple of 25's. Answer is D=19.