Summer is Coming! Join the Game of Timers Competition to Win Epic Prizes. Registration is Open. Game starts Mon July 1st.

 It is currently 18 Jul 2019, 06:31

### GMAT Club Daily Prep

#### 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

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

# If N is the product of all multiples of 10 between 199 and 301, what i

Author Message
TAGS:

### Hide Tags

Retired Moderator
Joined: 29 Apr 2015
Posts: 830
Location: Switzerland
Concentration: Economics, Finance
Schools: LBS MIF '19
WE: Asset Management (Investment Banking)
If N is the product of all multiples of 10 between 199 and 301, what i  [#permalink]

### Show Tags

26 Aug 2015, 12:23
16
00:00

Difficulty:

95% (hard)

Question Stats:

40% (01:50) correct 60% (02:12) wrong based on 233 sessions

### HideShow timer Statistics

If N is the product of all multiples of 10 between 199 and 301, what is the greatest integer m for which $$\frac{N}{10^{m}}$$ is an integer?

A. 10
B. 11
C. 13
D. 14
E. 15

What is wrong with my approach below?

_________________
Saving was yesterday, heat up the gmatclub.forum's sentiment by spending KUDOS!

PS Please send me PM if I do not respond to your question within 24 hours.
Manager
Joined: 01 Jan 2015
Posts: 62
Re: If N is the product of all multiples of 10 between 199 and 301, what i  [#permalink]

### Show Tags

26 Aug 2015, 13:40
6
3
reto wrote:
My approach: We need to find the number of trailing 0 of N, right? Ok, as learned it goes like this:

Between 199 and 301 there are 11 multiples of 10 ((200-300)/10 +1). So far so good. So N is the product of all multiples of 10 between 199 and 301. Basically this reads as follows:

N=(200*210*220*230...) which can be compressed by extracting a 10(20*21*22*23....). It follows that N=10^11* .... usually one can write down a factorial here but in this example, the multiples are not from 1 to something rather from 199 to ... so we can not use a factorial and calculate the trailing zeros from there. What would you offer?

How to solve it in an easy way without counting seperately? VeritasPrepKarishma (Would be glad to see your post).

Thanks

The product of multiples of 10 from 199 to 301 inclusive can be written as:(10*20)*(10*21)*(10*22)......(10*30). It can be rewritten as (10^11*30!)/19!.
30! has 7 trailing zeroes. 19! has 3 trailing zeroes. The numerator of (10^11*30!)/19! has 18 trailing zeroes and the denominator has 3 trailing zeroes. So, (10^11*30!)/19! will be some number with 15 trailing zeroes.
##### General Discussion
Retired Moderator
Joined: 29 Apr 2015
Posts: 830
Location: Switzerland
Concentration: Economics, Finance
Schools: LBS MIF '19
WE: Asset Management (Investment Banking)
Re: If N is the product of all multiples of 10 between 199 and 301, what i  [#permalink]

### Show Tags

26 Aug 2015, 12:30
My approach: We need to find the number of trailing 0 of N, right? Ok, as learned it goes like this:

Between 199 and 301 there are 11 multiples of 10 ((200-300)/10 +1). So far so good. So N is the product of all multiples of 10 between 199 and 301. Basically this reads as follows:

N=(200*210*220*230...) which can be compressed by extracting a 10(20*21*22*23....). It follows that N=10^11* .... usually one can write down a factorial here but in this example, the multiples are not from 1 to something rather from 199 to ... so we can not use a factorial and calculate the trailing zeros from there. What would you offer?

How to solve it in an easy way without counting seperately? VeritasPrepKarishma (Would be glad to see your post).

Thanks
_________________
Saving was yesterday, heat up the gmatclub.forum's sentiment by spending KUDOS!

PS Please send me PM if I do not respond to your question within 24 hours.
Retired Moderator
Joined: 29 Apr 2015
Posts: 830
Location: Switzerland
Concentration: Economics, Finance
Schools: LBS MIF '19
WE: Asset Management (Investment Banking)
Re: If N is the product of all multiples of 10 between 199 and 301, what i  [#permalink]

### Show Tags

26 Aug 2015, 12:37
reto wrote:
If N is the product of all multiples of 10 between 199 and 301, what is the greatest integer m for which is an integer?

A. 10
B. 11
C. 13
D. 14
E. 15

What is wrong with my approach below?

Here is the offical solution, ...

Count how many of the multiples of 10 have more than one 5:

200 = 52×8

250 = 52×10 = 53×2

300 = 52×12

One 5 of each of these is counted in our 11 multiples of 10, but there are 4 additional 5s, which can be used to make 4 additional 10s.

All in all, there are 11 + 4 = 15 10's in N. Thus, the greatest possible value of m is 15.
_________________
Saving was yesterday, heat up the gmatclub.forum's sentiment by spending KUDOS!

PS Please send me PM if I do not respond to your question within 24 hours.
CEO
Joined: 20 Mar 2014
Posts: 2620
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
GPA: 3.7
WE: Engineering (Aerospace and Defense)
Re: If N is the product of all multiples of 10 between 199 and 301, what i  [#permalink]

### Show Tags

26 Aug 2015, 13:41
reto wrote:
If N is the product of all multiples of 10 between 199 and 301, what is the greatest integer m for which is an integer?

A. 10
B. 11
C. 13
D. 14
E. 15

What is wrong with my approach below?

reto, the question has not been defined completely. "what is the greatest integer m for which ?? is an integer"?? Do you mind checking it again?
Retired Moderator
Joined: 29 Apr 2015
Posts: 830
Location: Switzerland
Concentration: Economics, Finance
Schools: LBS MIF '19
WE: Asset Management (Investment Banking)
Re: If N is the product of all multiples of 10 between 199 and 301, what i  [#permalink]

### Show Tags

28 Aug 2015, 10:19
Engr2012 wrote:
reto wrote:
If N is the product of all multiples of 10 between 199 and 301, what is the greatest integer m for which is an integer?

A. 10
B. 11
C. 13
D. 14
E. 15

What is wrong with my approach below?

reto, the question has not been defined completely. "what is the greatest integer m for which ?? is an integer"?? Do you mind checking it again?

Of course! Sorry for that. Happy to see your response!
_________________
Saving was yesterday, heat up the gmatclub.forum's sentiment by spending KUDOS!

PS Please send me PM if I do not respond to your question within 24 hours.
CEO
Joined: 20 Mar 2014
Posts: 2620
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
GPA: 3.7
WE: Engineering (Aerospace and Defense)
If N is the product of all multiples of 10 between 199 and 301, what i  [#permalink]

### Show Tags

28 Aug 2015, 10:34
3
2
reto wrote:
Engr2012 wrote:
reto wrote:
If N is the product of all multiples of 10 between 199 and 301, what is the greatest integer m for which is an integer?

A. 10
B. 11
C. 13
D. 14
E. 15

What is wrong with my approach below?

reto, the question has not been defined completely. "what is the greatest integer m for which ?? is an integer"?? Do you mind checking it again?

Of course! Sorry for that. Happy to see your response!

Sure, look below:

N = 200*210*220*230*240*250*260*270*280*290*300 , you need to find the value of integer m, such that N/$$10^m$$ is an integer. For these questions, whenever you need to find the max. or min m such that N/$$P^m$$ = integer, make sure to break P down in the form of its Prime factors.

After this, the determining value will be the prime factor with the lowest power.

Example, in this case, $$10^m = 2^m*5^m$$ and for the given series , 5s will be a lot more scarcer than 2s and thus power of 5 will determine how many $$10^m$$ can we get.

Once you establish which prime factor will be scarcer, it now comes down to counting how many 5s you have in N.

N = 200*210*220*230*240*250*260*270*280*290*300 = $$2^{11}$$*$$5^{11}$$*(20*21*22*23*24*25*26*27*28*29*30) and you have 4 5s in the remaining (20*21*22*23*24*25*26*27*28*29*30), bringing the total count of 5s to 15.

Thus N/10^m = integer for m = 15. E is the correct answer.

Although the above looks time consuming, it is actually not and I was able to get the answer in 1.2 minutes.

Hope this helps.
Manager
Joined: 14 Jul 2014
Posts: 164
Location: United States
Schools: Duke '20 (D)
GMAT 1: 600 Q48 V27
GMAT 2: 720 Q50 V37
GPA: 3.2
Re: If N is the product of all multiples of 10 between 199 and 301, what i  [#permalink]

### Show Tags

17 Jan 2016, 20:36
11 tens from 200, 210, 220... 300.

And, 4 tens from 4 fives in 200, 250 and 300.
Intern
Joined: 04 Feb 2017
Posts: 44
GMAT 1: 690 Q50 V34
Re: If N is the product of all multiples of 10 between 199 and 301, what i  [#permalink]

### Show Tags

20 Aug 2017, 11:40
Easy one..

200...to ..300...= 13 zeores..
+
250== there 2 5's are coming... so two more zeroes...

Non-Human User
Joined: 09 Sep 2013
Posts: 11694
Re: If N is the product of all multiples of 10 between 199 and 301, what i  [#permalink]

### Show Tags

25 Sep 2018, 14:31
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 N is the product of all multiples of 10 between 199 and 301, what i   [#permalink] 25 Sep 2018, 14:31
Display posts from previous: Sort by