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

It is currently 20 Jul 2019, 15:09

Close

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.

Close

Request Expert Reply

Confirm Cancel

What is the highest power of 7 in 5000! ?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 56304
What is the highest power of 7 in 5000! ?  [#permalink]

Show Tags

New post 29 Jan 2019, 03:55
00:00
A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

57% (02:21) correct 43% (01:58) wrong based on 72 sessions

HideShow timer Statistics


CEO
CEO
User avatar
D
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 2960
Location: India
GMAT: INSIGHT
Schools: Darden '21
WE: Education (Education)
Reviews Badge
What is the highest power of 7 in 5000! ?  [#permalink]

Show Tags

New post 29 Jan 2019, 04:03
1
Bunuel wrote:
What is the highest positive integer power of 7 in 5000!?

(A) 714
(B) 816
(C) 832
(D) 835
(E) 4998


What is the highest power of 7 in 5000!?


CONCEPT: Power of any Prime Number in any factorial can be calculated by following understanding

Power of prime x in\(n! = [n/x] + [n/x^2] + [n/x^3] + [n/x^4] +\)... and so on
Where,

[\(n/x\)] = No. of Integers that are multiple of x from 1 to n

[\(n/x^2\)] = No. of Integers that are multiple of x^2 from 1 to n whose first power has been counted in previous step and second is being counted at this step

[\(n/x^3\)] = No. of Integers that are multiple of x^3 from 1 to n whose first two powers have been counted in previous two step and third power is counted at this step
And so on.....


Where [\(n/x\)] is greatest Integer value of (n/x) less than or equal to (n/x)
i.e. [100/3] = [33.33] = 33
i.e. [100/9] = [11.11] = 11 etc.


i.e. Power of 7 in 5000! \(= [5000/7] + [5000/7^2] + [5000/7^3] + [5000/7^4] + ...\)


i.e. Power of 7 in 5000! \(= 714 + 102 + 14 + 2 = 832\)

Answer: Option C
_________________
Prosper!!!
GMATinsight
Bhoopendra Singh and Dr.Sushma Jha
e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772
Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi
http://www.GMATinsight.com/testimonials.html

ACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION
GMAT Club Legend
GMAT Club Legend
User avatar
D
Joined: 18 Aug 2017
Posts: 4252
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
GMAT ToolKit User Premium Member CAT Tests
Re: What is the highest power of 7 in 5000! ?  [#permalink]

Show Tags

New post 29 Jan 2019, 04:14
1
Bunuel wrote:
What is the highest positive integer power of 7 in 5000!?

(A) 714
(B) 816
(C) 832
(D) 835
(E) 4998


What is the highest power of 7 in 5000!?



5000!/7 + 5000/7^2 + 5000/7^3 + 50007^4

= 714+102+14+2
832
IMO C
_________________
If you liked my solution then please give Kudos. Kudos encourage active discussions.
Intern
Intern
avatar
B
Joined: 26 Dec 2018
Posts: 2
Re: What is the highest power of 7 in 5000! ?  [#permalink]

Show Tags

New post 29 Jan 2019, 22:49
Can someone explain this further? I'm not following the solutions here. Thanks!
VP
VP
avatar
G
Joined: 09 Mar 2018
Posts: 1002
Location: India
Re: What is the highest power of 7 in 5000! ?  [#permalink]

Show Tags

New post 29 Jan 2019, 23:27
[quote="Bunuel"]What is the highest positive integer power of 7 in 5000!?

(A) 714
(B) 816
(C) 832
(D) 835
(E) 4998


What is the highest power of 7 in 5000!?


This concept is tested here
https://gmatclub.com/forum/everything-a ... 85592.html

5000/7 + 5000/7^2 + 5000/7^3 + 5000/7^4

C

Posted from my mobile device
_________________
If you notice any discrepancy in my reasoning, please let me know. Lets improve together.

Quote which i can relate to.
Many of life's failures happen with people who do not realize how close they were to success when they gave up.
Math Expert
User avatar
V
Joined: 02 Aug 2009
Posts: 7764
Re: What is the highest power of 7 in 5000! ?  [#permalink]

Show Tags

New post 29 Jan 2019, 23:38
ccheng26 wrote:
Can someone explain this further? I'm not following the solutions here. Thanks!



OK so we are looking for maximum value of n when \(7^n<5000!\).
This basically means the number of times 7 comes in the product 1*2*3*4*...*5000.

Now when we divide by 7 what happens..

\(\frac{5000}{7}=714.xy\), so 714 gives us the number which have 7 in it.. 7, 14, 21, ......4998.
\(\frac{5000}{49}=102.04\)~120 gives us the number which are div by 49 as there is an extra 7 in these numbers... 49, 343....
\(\frac{5000}{49*7}=14.57\)~14 gives us the number which are div by 49*7 as there is an extra 7 in these numbers over and above what we have counted till now... 343....
similarly \(\frac{5000}{49*7*7}=2.08\), so 2 numbers more 2401 and 4802
Next 49*7*7*7 goes beyond 5000 and will give us an answer less than 1..

Now we have calculated all 7s whether as 49, 343 etc = 714+102+14+2=832
So our Ans 10+3+1=14
_________________
Intern
Intern
avatar
B
Joined: 06 Dec 2017
Posts: 14
GMAT ToolKit User CAT Tests
Re: What is the highest power of 7 in 5000! ?  [#permalink]

Show Tags

New post 27 Mar 2019, 01:45
1
While the concept has been explained in earlier posts,

\(\frac{5000}{7}\) = 714 ;


5000/7^2 = 5000/7 x 1/7 = 714/ 7 = 102 ; (714 = 5000/7 from previous step)

Similarly

5000/7^3 = 102/7 = 14 ;
5000/7^4 = 14/7 = 2;

Add 714+102+14+2 = 832
GMAT Club Bot
Re: What is the highest power of 7 in 5000! ?   [#permalink] 27 Mar 2019, 01:45
Display posts from previous: Sort by

What is the highest power of 7 in 5000! ?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne