BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

60!

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
Legendary Member
Posts: 510
Joined: Thu Aug 07, 2014 2:24 am
Thanked: 3 times
Followed by:5 members

60!

by j_shreyans » Mon Jul 13, 2015 8:18 am
If 60! is written out as an integer, with how many consecutive 0's will that integer end?

A)6

B)12

C)14

D)42

E)56

OAC
Top
Source: — Problem Solving |

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 16207
Joined: Mon Dec 08, 2008 6:26 pm
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1268 members
GMAT Score:770

by Brent@GMATPrepNow » Mon Jul 13, 2015 8:23 am
If 60! is written out as an integer, with how many consecutive 0's will that integer end?
A) 6
B) 12
C) 14
D) 42
E) 56
60! = (1)(2)(3)(4)(5)(6) . . . (57)(58)(59)(60)
For every PAIR of one 2 and one 5, we get a product of 10, which accounts for one zero at the end of the integer.
So, the question is "How many PAIR of one 2 and one 5 are "hiding" in the product?"

Well, there is no shortage of 2's hiding in the product. In fact, there are FAR MORE 2's than 5's. So, all we need to do is determine how many 5's are hiding in the product.
60! = (1)(2)(3)(4)(5)(6) . . . (57)(58)(59)(60)
= (1)(2)(3)(4)(5)(6)(7)(8)(9)(2)(5)(11)...(3)(5)...(4)(5)...(5)(5)...(6)(5)...(7)(5)...(8)(5)...(9)(5)...(2)(5)(5)...(11)(5)...(56)(57)(58)(59)(12)(5)
In total, there are 14 5's hiding in the product.
And there are MORE THAN 14 2's hiding in the product.

So, there are 14 pairs of 2's and 5'2, which means the integer ends with 14 zeros

Answer: C

If you'd like additional practice, here's a similar question: https://www.beatthegmat.com/integers-t270158.html

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
Image
Top
User avatar
MBA Admissions Consultant
Posts: 2279
Joined: Fri Nov 11, 2011 7:51 am
Location: New York
Thanked: 660 times
Followed by:266 members
GMAT Score:770

by Jim@StratusPrep » Mon Jul 13, 2015 8:25 am
Every time you have 5 x 2 you get a zero at the end of a number. This is also the ONLY time you will get a zero at the end. As there are many more 2s than there are 5s, the number of zeros will be equal to the number of prime factors of 5 that 60! has. Here are the counts:


5 - 1
10 - 1
15 - 1
20 - 1
25 - 2
30 - 1
35 - 1
40 - 1
45 - 1
50 - 2
55 - 1
60 - 1

Total them up to 14.
GMAT Answers provides a world class adaptive learning platform.
-- Push button course navigation to simplify planning
-- Daily assignments to fit your exam timeline
-- Organized review that is tailored based on your abiility
-- 1,000s of unique GMAT questions
-- 100s of handwritten 'digital flip books' for OG questions
-- 100% Free Trial and less than $20 per month after.
-- Free GMAT Quantitative Review

Image
Top
Legendary Member
Posts: 518
Joined: Tue May 12, 2015 8:25 pm
Thanked: 10 times

by nikhilgmat31 » Tue Jul 14, 2015 2:40 am
By mistake I counted as 12

60,50,40,30,20,10 = 6
55,45,35,25,15,5 = 6

But then realized than 25 has 2 - 5's & 50 has 2 - 5's so both has 2 extra 5's . counting total to 14.

Good Question.
Top
User avatar
GMAT Instructor
Posts: 15539
Joined: Tue May 25, 2010 12:04 pm
Location: New York, NY
Thanked: 13060 times
Followed by:1906 members
GMAT Score:790

by GMATGuruNY » Tue Jul 14, 2015 4:10 am
If 60! is written out as an integer, with how many consecutive 0's will that integer end?
6
12
14
42
56
60! = 60*59*58*....*3*2*1.

Since 10=2*5, EVERY COMBINATION OF 2*5 contained within the prime-factorization of 60! will yield a 0 at the end of the integer representation of 60!.
The prime-factorization of 60! is composed of FAR MORE 2'S than 5's.
Thus, the number of 0's depends on the NUMBER OF 5's contained within 60!.

To count the number of 5's, simply divide increasing POWERS OF 5 into 60.

Every multiple of 5 within 60! provides at least one 5:
60/5 = 12 --> twelves 5's.
Every multiple of 5² provides a SECOND 5:
60/5² = 2 --> two more 5's.
Thus, the total number of 5's contained within 60! = 12+2 = 14.

The correct answer is C.

Another example:

If 200! is written out as an integer, with how many consecutive 0's will that integer end?

Every multiple of 5 within 200! provides at least one 5:
200/5 = 40 --> forty 5's.
Every multiple of 5² provides a SECOND 5:
200/5² = 8 --> eight more 5's.
Every multiple of 5³ provides a THIRD 5:
200/5³ = 1 --> one more 5.
Thus, the total number of 5's contained within 200! = 40+8+1 = 49.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Top
Master | Next Rank: 500 Posts
Posts: 313
Joined: Tue Oct 13, 2015 7:01 am
Thanked: 2 times

by jain2016 » Wed Feb 17, 2016 9:18 am
Every multiple of 5 within 200! provides at least one 5:
200/5 = 40 --> forty 5's.
Every multiple of 5² provides a SECOND 5:
200/5² = 8 --> eight more 5's.
Every multiple of 5³ provides a THIRD 5:
200/5³ = 1 --> one more 5.
Thus, the total number of 5's contained within 200! = 40+8+1 = 49.
[/quote]

Hi Mitch ,

Can you please explain the above part.

Also 200/5³ is not perfectly divisible so how come ?

Please explain.

Many thanks in advance.

SJ
Top

GMAT/MBA Expert

User avatar
Elite Legendary Member
Posts: 10392
Joined: Sun Jun 23, 2013 6:38 pm
Location: Palo Alto, CA
Thanked: 2867 times
Followed by:511 members
GMAT Score:800

by [email protected] » Wed Feb 17, 2016 10:27 am
Hi jain2016,

Mitch created that 'table' to show you the number of terms that exist in 200!

200! has 40 multiples of 5 in it (5, 10, 15, 20.....195, 200). You can prove that point by calculating 200/5 = 40. That deduction would still hold true even if we were dealing with 201! though...

201/5 = 40 and some left over, so there are 40 multiples of 5 in 201!

When Mitch calculated 200/(5^3), that was meant to prove that there was just 1 multiple of 125 in the first 200 terms.

GMAT assassins aren't born, they're made,
Rich
Contact Rich at [email protected]
Image
Top
Master | Next Rank: 500 Posts
Posts: 313
Joined: Tue Oct 13, 2015 7:01 am
Thanked: 2 times

by jain2016 » Wed Feb 17, 2016 9:21 pm
Hi ,

Thanks for your reply.

One more thing can we also do 200/5^4 ?

Please advise .

Thanks,

SJ
Top
User avatar
Legendary Member
Posts: 2135
Joined: Mon Feb 03, 2014 9:26 am
Location: https://martymurraycoaching.com/
Thanked: 955 times
Followed by:140 members
GMAT Score:800

by MartyMurray » Thu Feb 18, 2016 5:34 am
jain2016 wrote:One more thing can we also do 200/5^4 ?
What does 25 x 25 equal?

Now, answer your own question.
Marty Murray
Perfect Scoring Tutor With Over a Decade of Experience
MartyMurrayCoaching.com
Contact me at [email protected] for a free consultation.
Top
Master | Next Rank: 500 Posts
Posts: 313
Joined: Tue Oct 13, 2015 7:01 am
Thanked: 2 times

by jain2016 » Thu Feb 18, 2016 9:05 am
What does 25 x 25 equal?

Now, answer your own question.
[/quote]

Hi Marty ,

25X25 = 625 and I got the my answer.

Thanks for the direction.

SJ
Top

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 16207
Joined: Mon Dec 08, 2008 6:26 pm
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1268 members
GMAT Score:770

by Brent@GMATPrepNow » Thu Feb 18, 2016 10:22 am
jain2016 wrote:
What does 25 x 25 equal?

Now, answer your own question.
Hi Marty ,

25X25 = 625 and I got the my answer.

Thanks for the direction.

SJ[/quote]

Aside: there's a nice quick tip for squaring ANY number ending in 5 (e.g., 25², 35², 105², etc) in your head. Here's a free video that covers the simple technique: https://www.gmatprepnow.com/module/gmat ... video/1024

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
Image
Top
Post new topic Post Reply
• Page 1 of 1