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If x is the smallest positive integer that is not prime and

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

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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New post 02 Nov 2014, 02:40
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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New post 10 Jul 2016, 07:27
I got answer as 51 and addition of factors as 72.
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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New post 12 Nov 2016, 12:22
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.

Answer: D.
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what is the sum of the factors of x? [#permalink]

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New post 12 Feb 2017, 05:08
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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New post 12 Feb 2017, 05:42
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. :)

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

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New post 13 Feb 2017, 04:09
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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New post 19 Feb 2017, 11:20
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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Re: If x is the smallest positive integer that is not prime and [#permalink]

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New post 19 Feb 2017, 11:34
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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New post 29 Jun 2017, 17:20
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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New post 29 Jun 2017, 20:53
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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New post 29 Jun 2017, 21:49
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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New post 16 Jul 2017, 03:19
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.

Answer: D.


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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New post 11 Jan 2018, 08:20
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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New post 08 Feb 2018, 12:09
Hi All,

50! = (1)(2)(3)(4)....(48)(49)(50)

50! is clearly divisible by LOTS of different integers. Besides the 50 integers listed above, ANY integer that can be created by multiplying any of those values (or their factors) will divide into 50!

For example:
(2) and (49) are both factors of 50!, so (2)(49) = 98 will ALSO be a factor of 50!.

The question asks us to find the SMALLEST positive integer that is NOT PRIME and NOT a factor of 50! Once we have that value, we have to figure out the sum of its factors.

Rather than try to list out all of the possible factors of 50!, we have to think about the type of number that is NOT a factor of 50! If we think about prime-factorization, we'll find that the smallest prime factor in 50! is 2 (it can be found in ALL of the even integers). Similarly, we can find '3' in all of the multiples of 3 (3, 6, 9, etc.) and we can find '5' in all of the multiples of 5 (5, 10, 15, etc.). The largest prime factor in 50! is 47; to find a number that is NOT a factor of 50!, we have to find one that includes a prime factor that is NOT a factor of 50!

The next largest prime number above 47 is 53. 53 is NOT a factor of 50! and we cannot 'get to' 53 by multiplying any of the factors of 50! The prompt tells us that we need a number that is NOT prime though - the smallest number that has 53 as a factor and is NOT prime is 106 (which is 2x53). THAT is the smallest number that fits all of the restrictions that we're given.

From there, we know two of the factors of 106 already: 1 and 106, so the sum has to be GREATER THAN 106. The only answer among the 5 choices that makes any sense is....

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Re: If x is the smallest positive integer that is not prime and   [#permalink] 08 Feb 2018, 12:09

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