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# How many prime factors 100! has?

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Senior Manager
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How many prime factors 100! has? [#permalink]

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02 Dec 2010, 09:34
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Difficulty:

65% (hard)

Question Stats:

48% (01:10) correct 52% (01:20) wrong based on 64 sessions

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How many prime factors 100! has?

24
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31

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Re: How many prime factors 100! has? [#permalink]

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02 Dec 2010, 09:42
shrive555 wrote:
How many prime factors 100! has

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25
27
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100!=1*2*3*...*100, so 100! has all prime factors which are less than 100 and only them. It happens to be that there are 25 primes less than 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.

I really doubt that this is a realistic GMAT question: either you should remember the fact that there are 25 primes less than 100 or you should jut use brute force to count them.
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Re: How many prime factors 100! has? [#permalink]

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02 Dec 2010, 11:57
you're right B. i was going through the topic " finding the Number of powers of a prime number P in the n!" in Number Theory article and thought that we could find Number of prime factors in any factorial given through the formula n/p + n/p^2 ........till p^x < n
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Re: How many prime factors 100! has? [#permalink]

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03 Dec 2010, 02:56
Bunuel wrote:
shrive555 wrote:
How many prime factors 100! has

24
25
27
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31

100!=1*2*3*...*100, so 100! has all prime factors which are less than 100 and only them. It happens to be that there are 25 primes less than 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.

I really doubt that this is a realistic GMAT question: either you should remember the fact that there are 25 primes less than 100 or you should jut use brute force to count them.

Brunel, can we do this for every number?

for example, can we say that 55! would have as many prime number as the count of total prime numbers between 1 and 55?
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Re: How many prime factors 100! has? [#permalink]

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03 Dec 2010, 03:03
krishnasty wrote:
Bunuel wrote:
shrive555 wrote:
How many prime factors 100! has

24
25
27
29
31

100!=1*2*3*...*100, so 100! has all prime factors which are less than 100 and only them. It happens to be that there are 25 primes less than 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.

I really doubt that this is a realistic GMAT question: either you should remember the fact that there are 25 primes less than 100 or you should jut use brute force to count them.

Brunel, can we do this for every number?

for example, can we say that 55! would have as many prime number as the count of total prime numbers between 1 and 55?

Sure. 55!=1*2*3*...*55.

Now, as prime number cannot be written as a product of two factors, both of which are greater than 1 then prime more than 55 can not be a factor of 55! (prime more than 55, for example 71, just no way can occur in 55!).

Hope it's clear.
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Re: How many prime factors 100! has? [#permalink]

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12 Sep 2017, 01:25
shrive555 wrote:
How many prime factors 100! has?

24
25
27
29
31

We know that there are 25 prime numbers between 1 to 100 => when 100! is written => 100x99x98x97..........x3x2x1
they will get repeated so 25 is the number of primes

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Re: How many prime factors 100! has?   [#permalink] 12 Sep 2017, 01:25
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