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# What is the highest power of 7 in 5000! ?

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

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29 Jan 2019, 03:55
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65% (hard)

Question Stats:

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

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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!?

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What is the highest power of 7 in 5000! ?  [#permalink]

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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! $$= 714 + 102 + 14 + 2 = 832$$

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Re: What is the highest power of 7 in 5000! ?  [#permalink]

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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
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Re: What is the highest power of 7 in 5000! ?  [#permalink]

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29 Jan 2019, 22:49
Can someone explain this further? I'm not following the solutions here. Thanks!
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Re: What is the highest power of 7 in 5000! ?  [#permalink]

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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

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Posts: 7764
Re: What is the highest power of 7 in 5000! ?  [#permalink]

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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
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Re: What is the highest power of 7 in 5000! ?  [#permalink]

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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;

Re: What is the highest power of 7 in 5000! ?   [#permalink] 27 Mar 2019, 01:45
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