60!

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j_shreyans: Legendary Member; Posts: 510; Joined: Thu Aug 07, 2014 2:24 amFollowed by: 5 members; Thanked: 3 times

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

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Source: Beat The GMAT — Problem Solving | Mon Jul 13, 2015 8:18 am

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; 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

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Jim@StratusPrep: MBA Admissions Consultant; Posts: 2279; Joined: Fri Nov 11, 2011 7:51 amFollowed by: 266 members; Location: New York; Thanked: 660 times; 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.

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nikhilgmat31: 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.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; 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 5Â² provides a SECOND 5:
60/5Â² = 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 5Â² provides a SECOND 5:
200/5Â² = 8 --> eight more 5's.
Every multiple of 5Â³ provides a THIRD 5:
200/5Â³ = 1 --> one more 5.
Thus, the total number of 5's contained within 200! = 40+8+1 = 49.

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jain2016: 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 5Â² provides a SECOND 5:
200/5Â² = 8 --> eight more 5's.
Every multiple of 5Â³ provides a THIRD 5:
200/5Â³ = 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 200/5Â³ is not perfectly divisible so how come ?

Please explain.

Many thanks in advance.

SJ

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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.

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jain2016: 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

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MartyMurray: Legendary Member; Posts: 2135; Joined: Mon Feb 03, 2014 9:26 amFollowed by: 140 members; Location: https://martymurraycoaching.com/; Thanked: 955 times; 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.

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jain2016: 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

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; 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., 25Â², 35Â², 105Â², 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

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