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If x is the smallest positive integer that is not prime and not a fact  [#permalink]

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1
If x is the smallest positive integer that is not prime and not a factor of 50!, what is the sum of the factors of x?

1. 51
2. 54
3. 72
4. 162
5. 50!+2
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Re: If x is the smallest positive integer that is not prime and not a fact  [#permalink]

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50! is a very large number, but it will not have as factors any primes larger than 50.
The first prime larger than 50 is 53, but as x is not a prime lets consider 2x53 = 106
106 factors to 1x2x53 so sum of factors is 1+2+53+106 = 162

That gives D.
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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I was so wrong.
If there is a category "795+", this qualifies for it.
Good Learning in this !!
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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Extremely helpful Bunnuel. Did not understand VeritasPrep's explanation, but yours is on point.
Thanks!

Bunuel wrote:
brianlange77 wrote:
If x is the smallest positive integer that is not prime and not a factor of 50!, what is the sum of the factors of x?

A. 51
B. 54
C. 72
D. 162
E. 50!+2

Thanks for posting -- would be great for you to post the source of the question for easier reference in the future. That said, the factors of 50! are all the numbers from 1-50. The smallest positive integer that is not prime and not a factor of 50! is therefore 51. The factors of 51 are 1, 3, 17, and 51. The sum of those four numbers is 72.

Thoughts?

-Brian

In fact 51=3*17 IS a factor of 50!.

The smallest positive integer that is NOT prime and NOT a factor of 50! is 106=2*53. The sum of the factors of 106 is 162.

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what is the sum of the factors of x?  [#permalink]

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If x is the smallest positive integer that is not prime and not a factor of 50!, what is the sum of the factors of x?

ans given is 162,but it has to be 53 ans 1 i.e. 54
pls explain
thnx
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Re: what is the sum of the factors of x?  [#permalink]

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gupta87 wrote:
If x is the smallest positive integer that is not prime and not a factor of 50!, what is the sum of the factors of x?

ans given is 162,but it has to be 53 ans 1 i.e. 54
pls explain
thnx

Hey,

I presume you did not read the question carefully. x is the smallest positive integer which is not prime.

So tell me, can x be 53? Let me know if you still cannot figure out how to approach this. But you should try it again, keeping the above point in mind. Thanks,
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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gupta87 wrote:
If x is the smallest positive integer that is not prime and not a factor of 50!, what is the sum of the factors of x?

ans given is 162,but it has to be 53 ans 1 i.e. 54
pls explain
thnx

X should not be prime , so 53 is incorrect value for x ...
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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Bunuel wrote:
gupta87 wrote:
If x is the smallest positive integer that is not prime and not a factor of 50!, what is the sum of the factors of x?

ans given is 162,but it has to be 53 ans 1 i.e. 54
pls explain
thnx

Merging topics. Please refer to the discussion on page 1.

what is the level of this problem? more than 740?
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Joined: 02 Sep 2009
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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romych007 wrote:
Bunuel wrote:
gupta87 wrote:
If x is the smallest positive integer that is not prime and not a factor of 50!, what is the sum of the factors of x?

ans given is 162,but it has to be 53 ans 1 i.e. 54
pls explain
thnx

Merging topics. Please refer to the discussion on page 1.

what is the level of this problem? more than 740?

It's 700+. Hard to be more precise.
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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balentina1 wrote:
Factors of 50!: 1,2,3,......,50

If x is not a factor of 50! then one of his primes must be bigger than 49 ( the largest prime of 50!).

The next prime is 53.Thus x must be a multiple of 53.

Since x is not a prime the smallest possible multiple of 53 is 53 * 2 = 106

Sum of his factors: 1+2+53+106 = 162

Hope i helped This reasoning is fallacious because the statement in red ("If x is not a factor of 50! then one of his primes must be bigger than 49") is not true. For example, the numbers 47^2, 11^5, and 2^(48) are not factors of 50!, but none of them have prime factors bigger than 49.
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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SravnaTestPrep wrote:
The first number after n that will not be a factor of n! is the first prime after n. If the number cannot be a prime, then the first number that will not be a factor of n! is the p*2, where p is the first prime after n.

So in our case x will be 53*2=106 and the sum of the factors of 106 is 162.

The statement in red is not true for all positive integers n; it holds only when n is at least 6. For example, the smallest composite that is not a factor of 2! is 4, not 2*3 = 6; the smallest composite that is not a factor of 3! is also 4, not 2*5 = 10; the smallest composite that is not a factor of 4! is 9, not 2*5 = 10; and the smallest composite that is not a factor of 5! is again 9, not 2*7 =14.
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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mitharvardtutor wrote:
SravnaTestPrep wrote:
The first number after n that will not be a factor of n! is the first prime after n. If the number cannot be a prime, then the first number that will not be a factor of n! is the p*2, where p is the first prime after n.

So in our case x will be 53*2=106 and the sum of the factors of 106 is 162.

The statement in red is not true for all positive integers n; it holds only when n is at least 6. For example, the smallest composite that is not a factor of 2! is 4, not 2*3 = 6; the smallest composite that is not a factor of 3! is also 4, not 2*5 = 10; the smallest composite that is not a factor of 4! is 9, not 2*5 = 10; and the smallest composite that is not a factor of 5! is again 9, not 2*7 =14.

Yes , You are right. The idea is p*2 gives the smallest value after n, both p and 2 being primes. But in the case of 2!, 3!, 4!,5!, the smallest value after n is 2*2, 2*2, 3*3 and 3*3 with 2 and 3 being primes . Thanks for the correction.
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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Bunuel wrote:
brianlange77 wrote:
If x is the smallest positive integer that is not prime and not a factor of 50!, what is the sum of the factors of x?

A. 51
B. 54
C. 72
D. 162
E. 50!+2

Thanks for posting -- would be great for you to post the source of the question for easier reference in the future. That said, the factors of 50! are all the numbers from 1-50. The smallest positive integer that is not prime and not a factor of 50! is therefore 51. The factors of 51 are 1, 3, 17, and 51. The sum of those four numbers is 72.

Thoughts?

-Brian

In fact 51=3*17 IS a factor of 50!.

The smallest positive integer that is NOT prime and NOT a factor of 50! is 106=2*53. The sum of the factors of 106 is 162.

How do we even get to know that of the number 106? What's the underlying method?
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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TheMastermind wrote:
How do we even get to know that of the number 106? What's the underlying method?

Hi TheMastermind ,

We need to find out the number which doesn't have its factors within 50!.

The smallest such number should be a prime number > 50 = 53.

So, 53 will never divide 50! evenly. But we need to find out the non prime number as well. Let's multiply 53 with 2, you will get 106 as the smallest such number.

Now, I know you must be thinking why don't we have any other number between 53 and 106.

My friend, take any non prime number. You will get all the factors existing in 50!.

The rule is : The smallest such number is the next prime. If they are asking non prime, the answer should be even multiple of that prime.

I hope that makes sense. _________________
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If x is the smallest positive integer that is not prime and  [#permalink]

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smallest integer x which is not prime and factor of 50! must be = (2 * smallest prime number greater than 50) = (2 * 53) = 106
factors of 106 = {1, 2, 53, 106} => sum of factors = 162 => (D)
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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manasishah wrote:
If x is the smallest positive integer that is not prime and not a factor of 50!, what is the sum of the factors of x?

A. 51
B. 54
C. 72
D. 162
E. 50!+2

Factors of 50! include 1,2,3,.....50, hence x is greater than 50 & not a prime.

Our constraints are
- not a factor of 50!
- not a prime

Smallest prime greater than 50 is 53

Smallest composite number that can be formed with 53 as one of the factors is 53 * 2 = 106

Hence x = 106, factors of x are 1, 2, 53, 106

Sum of factors = 1+2+53+106 = 162

Thanks,
GyM
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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manasishah wrote:
If x is the smallest positive integer that is not prime and not a factor of 50!, what is the sum of the factors of x?

A. 51
B. 54
C. 72
D. 162
E. 50!+2

The smallest positive integer that is not a factor of 50! is 53 since 53 is a prime. However, since we don’t want the number to be a prime, we can multiply 53 by 2 and obtain 106, and 106 will be the smallest positive integer that is not prime and not a factor of 50!.

The sum of the factors of 106 is 106 + 1 + 53 + 2 = 162.

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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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If the factors of 51 (3 and 17) are also factors of 50, doesn't that imply that the factor 2 of 106 (2 and 53), is also a factor of 50?
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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salphonso wrote:
If the factors of 51 (3 and 17) are also factors of 50, doesn't that imply that the factor 2 of 106 (2 and 53), is also a factor of 50?

Hi salphonso,

50! = (50)(49)(48)....(3)(2)(1)

Thus, any of those 50 numbers is a factor of 50! AND any number that can be created by multiplying any combination of those numbers together (or any of the factors of those numbers together) will ALSO be a factor.

Thus, 51 IS a factor of 50! because 51 = (3)(17).... and both 3 and 17 are factors of 50!

106 = (2)(53).... but while 2 is a factor of 50!, 53 is NOT a factor. It is not among those 50 integers and there is no way to get to 53 by multiplying any of those numbers (or their factors) together. The smallest number that is NOT prime and NOT a factor of 50! is 106.

GMAT assassins aren't born, they're made,
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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EMPOWERgmatRichC wrote:
salphonso wrote:
If the factors of 51 (3 and 17) are also factors of 50, doesn't that imply that the factor 2 of 106 (2 and 53), is also a factor of 50?

Hi salphonso,

50! = (50)(49)(48)....(3)(2)(1)

Thus, any of those 50 numbers is a factor of 50! AND any number that can be created by multiplying any combination of those numbers together (or any of the factors of those numbers together) will ALSO be a factor.

Thus, 51 IS a factor of 50! because 51 = (3)(17).... and both 3 and 17 are factors of 50!

106 = (2)(53).... but while 2 is a factor of 50!, 53 is NOT a factor. It is not among those 50 integers and there is no way to get to 53 by multiplying any of those numbers (or their factors) together. The smallest number that is NOT prime and NOT a factor of 50! is 106.

GMAT assassins aren't born, they're made,
Rich

So we're looking at pairs that are factors, got it. Thank you! Re: If x is the smallest positive integer that is not prime and   [#permalink] 28 Jul 2020, 03:59

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