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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  [#permalink]

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30 Jul 2013, 11:07
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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?

A. 51
B. 54
C. 72
D. 162
E. 50!+2
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Re: Doubt: GMAT Number Theory Question  [#permalink]

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30 Jul 2013, 23:47
16
9
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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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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08 Feb 2018, 11:09
5
2
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]

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12 Dec 2013, 04:50
6
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
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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31 Jul 2013, 05:34
1
brianlange77 wrote:
Just for the sake of debate --- explain to me again how the number 51 (not its factors, but the number itself) is a factor of 50!?
Thanks.
-Brian

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?

If $$x=51$$, then $$x$$ is a factor of $$50!$$, $$\frac{50!}{51}=integer$$. I do not get what you mean by "not its factors, but the number itself"...

Do you mean something like $$50!=1*2*4*...*16*...*18*...*50$$[all numbers except 3 and 17]$$*51$$ ? (so you see the 51 in the number 50! ?)
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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31 Jul 2013, 05:38
2
Ahh... total mis-read on my part. Good work!
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Re: Doubt: GMAT Number Theory Question  [#permalink]

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31 Jul 2013, 13:20
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.

Bunuel,

How did you know it was 106=53*2? What is the thought process.

53 is a prime a number, obviously not in 50!. 53 cannot be the answer because the number must not be a factor of 50! and not a prime. How do you know for sure that there are not any numbers between 51 and 106 that might satisfy the condition?
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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31 Jul 2013, 16:59
2
1
I'll start the answer process for you....

-51 has factors of 3 and 17, so that's out, because those are both factors of 50!
-52 has factors of 2 (twice) and 13, so that's out...
-53 is not a factor of 50!, but it's prime
-But, 106 (which is 2*53) is the next smallest choice (any other non-factors between 53 and 106 would also be prime, violating the terms of the problem.)

Thoughts?
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Re: Doubt: GMAT Number Theory Question  [#permalink]

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01 Aug 2013, 01:27
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.

Yes Bunuel you absolutely correct the answer should be D...
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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01 Aug 2013, 10:01
brianlange77 wrote:
I'll start the answer process for you....

-51 has factors of 3 and 17, so that's out, because those are both factors of 50!
-52 has factors of 2 (twice) and 13, so that's out...
-53 is not a factor of 50!, but it's prime
-But, 106 (which is 2*53) is the next smallest choice (any other non-factors between 53 and 106 would also be prime, violating the terms of the problem.)

Thoughts?

Hi Brain,

In this case 106 which is 2*53 but 2 is a factor of 50! as in the previous case of 51 was out because 3 and 17 were factors of 50!.

can u explain y answer is 106 even 2 is in factor of 50!?

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

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01 Aug 2013, 18:26
rrsnathan wrote:
brianlange77 wrote:
I'll start the answer process for you....

-51 has factors of 3 and 17, so that's out, because those are both factors of 50!
-52 has factors of 2 (twice) and 13, so that's out...
-53 is not a factor of 50!, but it's prime
-But, 106 (which is 2*53) is the next smallest choice (any other non-factors between 53 and 106 would also be prime, violating the terms of the problem.)

Thoughts?

Hi Brain,

In this case 106 which is 2*53 but 2 is a factor of 50! as in the previous case of 51 was out because 3 and 17 were factors of 50!.

can u explain y answer is 106 even 2 is in factor of 50!?

regards,
RRSNATHAN

The number in discussion is 106. We need to see if that number is a factor of 50! or not.
We need not worry about 2 being a factor right...

Eg: (100/14) => 50*2/7*2
Does not imply 14 is a factor of 100...
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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20 Aug 2013, 22:34
avinashrao9 wrote:
rrsnathan wrote:
brianlange77 wrote:
I'll start the answer process for you....

-51 has factors of 3 and 17, so that's out, because those are both factors of 50!
-52 has factors of 2 (twice) and 13, so that's out...
-53 is not a factor of 50!, but it's prime
-But, 106 (which is 2*53) is the next smallest choice (any other non-factors between 53 and 106 would also be prime, violating the terms of the problem.)

Thoughts?

Hi Brain,

In this case 106 which is 2*53 but 2 is a factor of 50! as in the previous case of 51 was out because 3 and 17 were factors of 50!.

can u explain y answer is 106 even 2 is in factor of 50!?

regards,
RRSNATHAN

The number in discussion is 106. We need to see if that number is a factor of 50! or not.
We need not worry about 2 being a factor right...

Eg: (100/14) => 50*2/7*2
Does not imply 14 is a factor of 100...

Hi anybody pls explain this

Still i am not clear in this.

In explanation it is mentioned as
-51 has factors of 3 and 17, so that's out, because those are both factors of 50!
Here its mentioned as 3 and 17 are the factors of 59!.

In the same way 106 has two factors 2 and 53 in this 2 is the factor of 50! or not??

plz explain this.

-Rrsnathan
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Posts: 52288
Re: If x is the smallest positive integer that is not prime and  [#permalink]

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21 Aug 2013, 01:36
rrsnathan wrote:
avinashrao9 wrote:
rrsnathan wrote:
Hi Brain,

In this case 106 which is 2*53 but 2 is a factor of 50! as in the previous case of 51 was out because 3 and 17 were factors of 50!.

can u explain y answer is 106 even 2 is in factor of 50!?

regards,
RRSNATHAN

The number in discussion is 106. We need to see if that number is a factor of 50! or not.
We need not worry about 2 being a factor right...

Eg: (100/14) => 50*2/7*2
Does not imply 14 is a factor of 100...

Hi anybody pls explain this

Still i am not clear in this.

In explanation it is mentioned as
-51 has factors of 3 and 17, so that's out, because those are both factors of 50!
Here its mentioned as 3 and 17 are the factors of 59!.

In the same way 106 has two factors 2 and 53 in this 2 is the factor of 50! or not??

plz explain this.

-Rrsnathan

x is the smallest positive integer that is not prime and not a factor of 50! --> x=106 --> 106 is NOT a prime and is NOT 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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Updated on: 21 Aug 2013, 02:59
Hi,

Thanks for the explanation.

106 is NOT a prime and is NOT a factor of 50!
if this ur explanation

Then 51! is also not factor of 50! and not prime too. For omitting this we had an explanation as " 51 has factors of 3 and 17, so that's out, because those are both factors of 50!"

The same explanation will go for 106 - as 106 has factors 2 and 53, 2 is factors of 50! right?

Rrsnathan.

Originally posted by rrsnathan on 21 Aug 2013, 02:52.
Last edited by rrsnathan on 21 Aug 2013, 02:59, edited 1 time in total.
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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21 Aug 2013, 02:59
1
rrsnathan wrote:
Hi,

Thanks for the explanation.

106 is NOT a prime and is NOT a factor of 50!
if this ur explanation

Then 51! is also not factor of 50! and not prime too. For omitting this we had an explanation as " 51 has factors of 3 and 17, so that's out, because those are both factors of 50!"

The same explanation will go for 106 - as 106 has factors 2 and 53, both are in factors of 50! right?

Rrsnathan.

No. 53 is NOT 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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21 Aug 2013, 03:06
Bunuel wrote:
rrsnathan wrote:
Hi,

Thanks for the explanation.

106 is NOT a prime and is NOT a factor of 50!
if this ur explanation

Then 51! is also not factor of 50! and not prime too. For omitting this we had an explanation as " 51 has factors of 3 and 17, so that's out, because those are both factors of 50!"

The same explanation will go for 106 - as 106 has factors 2 and 53, both are in factors of 50! right?

Rrsnathan.

No. 53 is NOT a factor of 50!

Perfectly understood.

Thanks a lot Bunuel.
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If x is the smallest positive integer that is not prime and  [#permalink]

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01 Aug 2014, 03:38
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

This question has been discussed before, but I could not understand something from Bunuel's reply.
he said
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.

I understand that 53 is not factor of 50! since it is prime,, so multiplying by 2 make 53 not prime and not factor of 50! as well.
But, can someone explain how we can tell that none between 53 and 106 is not factor of 50!?
Is it becuase only prime numbers greater than 50 is not factors of 50!
so the smallest prime greater than 50 is 53 and thus 106 is smallest ineter that is NOT prime and NOT a factor of 50!?

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

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01 Aug 2014, 05:54
2
sehosayho 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

This question has been discussed before, but I could not understand something from Bunuel's reply.
he said
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.

I understand that 53 is not factor of 50! since it is prime,, so multiplying by 2 make 53 not prime and not factor of 50! as well.
But, can someone explain how we can tell that none between 53 and 106 is not factor of 50!?
Is it becuase only prime numbers greater than 50 is not factors of 50!
so the smallest prime greater than 50 is 53 and thus 106 is smallest ineter that is NOT prime and NOT a factor of 50!?

All numbers between 50 and 106, are either primes (and we know that x is NOT a prime) or factors of 50!, because it has all their primes in higher powers. For example:

59 is a prime, hence x cannot be 59.
60 = 2^2*3*5. 50! will for sure have 2, 3, and 5 in higher power than 2, 1 and 1, respectively.

Hope it's clear.

P.S. Please do not double post a topic, put your questions in existing one. Thank you.
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If x is the smallest positive integer that is not prime and  [#permalink]

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02 Aug 2014, 16:54
4
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.
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Re: If x is the smallest positive integer that is not prime and  [#permalink]

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02 Aug 2014, 22:20
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

D is the answer. Thanks for the explanation. It is difficult to understand the question when you first read it though.
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Re: If x is the smallest positive integer that is not prime and &nbs [#permalink] 02 Aug 2014, 22:20

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