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# Number of 5's in 100! =?

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Intern
Joined: 15 Apr 2017
Posts: 43
Number of 5's in 100! =?  [#permalink]

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02 Nov 2017, 13:26
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Can someone help with the strategy to solve such questions like this?

Number of 5's in 100! =?

Number of 2s in 200! =?

Number of 14 in 100! =?

Number of 28 in 100! =?

Number of 100 in 200! =?

@bruel

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Senior Manager
Joined: 13 Oct 2016
Posts: 288
GMAT 1: 600 Q44 V28
Number of 5's in 100! =?  [#permalink]

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02 Nov 2017, 14:40
1
The best strategy which i have come across is :

The highest power of prime number p in n! = [n/p1] + [n/p2] + [n/p3] + [n/p4] + …
[x] denotes the greatest integer less than or equal to x.
[1.2] =1
[4] = 4

For example: Find the highest power of 3 in 100!

= [100/3] + [100/32] + [100/33] + [100/34] + [100/35] + …
= 33 + 11 + 3 + 1 + 0 (from here on greatest integer function evaluates to zero)
= 48.

Now coming back to your first problem:
Number of 5's in 100! =?

[100/5] + [100/(5^2)] = 20+4=24

For no: which is non prime like 24,40 etc. use the below mentioned methodology with an example:

We learnt before that any non prime number can be expressed as a product of its prime factors.

We will use this concept to solve this type of questions.

What is the greatest power of 30 in 50!

30 = 2 * 3 * 5.

Find what the highest power of each prime is in the given factorial

We have 25 + 12 + 6 + 3 + 1 = 47 2s in 50!
16 + 5 + 1 = 22 3s in 50!
10 + 2 = 12 5s in 50!

We need a combination of one 2, one 3 and one 5 to get 30. In the given factorial we have only 12 5s. Hence only twelve 30s are possible. So the greatest power of 30 in 50! is 12.

Now let's solve the 3 rd problem which is Number of 14 in 100! =?

14=2*7

We have 50+ 25 + 12 + 6 + 3 + 1 = 97 2s

We have 14+2 = 16 7s

We need a combination of one 2, one 7 to get 14. In the given factorial we have only 16 7s. Hence only sixteen 14s are possible. So the greatest power of 7 in 100! is 16.

--== Message from the GMAT Club Team ==--

THERE IS LIKELY A BETTER DISCUSSION OF THIS EXACT QUESTION.
This discussion does not meet community quality standards. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.

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Number of 5's in 100! =? &nbs [#permalink] 02 Nov 2017, 14:40
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