Last visit was: 21 Jun 2024, 07:07 It is currently 21 Jun 2024, 07:07
Toolkit
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
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.
Request Expert Reply

# If the number 200! is written in the form p × 10^q, where p and q are

SORT BY:
Tags:
Show Tags
Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 93944
Own Kudos [?]: 633615 [34]
Given Kudos: 82411
Most Helpful Reply
GMAT Club Legend
Joined: 08 Jul 2010
Status:GMAT/GRE Tutor l Admission Consultant l On-Demand Course creator
Posts: 6016
Own Kudos [?]: 13650 [11]
Given Kudos: 125
Location: India
GMAT: QUANT+DI EXPERT
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
WE:Education (Education)
General Discussion
Intern
Joined: 04 May 2014
Posts: 27
Own Kudos [?]: 48 [4]
Given Kudos: 13
Math Expert
Joined: 02 Sep 2009
Posts: 93944
Own Kudos [?]: 633615 [2]
Given Kudos: 82411
Re: If the number 200! is written in the form p × 10^q, where p and q are [#permalink]
1
Kudos
1
Bookmarks
Expert Reply
Bunuel wrote:
If the number 200! is written in the form p × 10^q, where p and q are integers, what is the maximum possible value of q?

(A) 40
(B) 48
(C) 49
(D) 55
(E) 64

MANHATTAN GMAT OFFICIAL SOLUTION:

To maximize the value of q, we need to constrain p to NOT be a multiple of 10. In other words, q should be the count of every factor of 10 in 200!, and p would simply be the product of the remaining factors of 200!.

To count factors of 10 in 200!, we could start by counting the multiples of 10 between 1 and 200, inclusive. But this method would undercount the number of 10's that are factors of 200!. For example:

200! has 2 and 5 as factors, which multiply to 10, and thus another factor of 10.
200! has 6 and 15 as factors, which multiply to 90, and thus another factor of 10.
200! has 8 and 125 as factors, which multiply to 1000, and thus three more factors of 10.

Therefore, this problem is really about counting the number of 2 × 5 factor pairs that can be made from the factors of 200.

Let's count the number of 5s found among the factors of 200!:
Attachment:

2015-06-08_1557.png [ 51.52 KiB | Viewed 11770 times ]

We have counted not only the multiples of 5, but also the multiples of 25 and 125, as these higher multiples contribute more than one factor of 5 to the total.

There are eight multiples of 25 in our range (namely, 25, 50, 75, 100, 125, 150, 175, and 200). Each of these are also multiples of 5, so we have already counted one factor of 5 for them, and thus each of the eight multiples of 25 contributes one additional factor of 5. Finally, 125 contributes a total of three 5s to our count — but we already counted one factor of 5 when we counted multiples of 5, and another when we counted multiples of 25, leaving only one additional factor of 5 that we need to count for 125.

Thus, the prime factorization for 200! includes 40 + 8 + 1 = 49 factors of 5.

It is fairly easy to see that there are many more factors of 2 in 200!, as there are 100 even numbers between 1 and 200, not to mention the additional 2's coming from the multiples of 4, 8, 16, 32, etc. that contribute extra factors of 2.

Thus, we can only make 49 of the 2 × 5 factor pairs, and the maximum value of q is 49.

The correct answer is C.
Math Expert
Joined: 02 Sep 2009
Posts: 93944
Own Kudos [?]: 633615 [1]
Given Kudos: 82411
Re: If the number 200! is written in the form p × 10^q, where p and q are [#permalink]
1
Bookmarks
Expert Reply
Bunuel wrote:
If the number 200! is written in the form p × 10^q, where p and q are integers, what is the maximum possible value of q?

(A) 40
(B) 48
(C) 49
(D) 55
(E) 64

For similiar questions check Trailing Zeros Questions and Power of a number in a factorial questions in our Special Questions Directory.

Theory on Trailing Zeros: everything-about-factorials-on-the-gmat-85592.html
Intern
Joined: 27 Jul 2017
Posts: 37
Own Kudos [?]: 11 [1]
Given Kudos: 48
Re: If the number 200! is written in the form p × 10^q, where p and q are [#permalink]
1
Kudos
This is a trailing zero concept question: https://gmatclub.com/forum/everything-a ... 85592.html

# of trailing zeros 200! has is 49. Therefore, the answer is C.
Non-Human User
Joined: 09 Sep 2013
Posts: 33645
Own Kudos [?]: 839 [0]
Given Kudos: 0
Re: If the number 200! is written in the form p × 10^q, where p and q are [#permalink]
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 the number 200! is written in the form p × 10^q, where p and q are [#permalink]
Moderator:
Math Expert
93944 posts