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

Question

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

Bunuel · Accepted Answer

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.

EMPOWERgmatRichC · Answer

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....

Final Answer:  D

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

brianlange77 · Answer

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?

balentina1 · Answer

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

sehosayho · Answer

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!?

Let me know please.

Bunuel · Answer

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.

SVaidyaraman · Answer

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.

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

gupta87 · Answer

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

EgmatQuantExpert · Answer

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, Saquib Quant Expert e-GMAT Register for our Free Session on Number Properties (held every 3rd week) to solve exciting 700+ Level Questions in a classroom environment under the real-time guidance of our Experts

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ScottTargetTestPrep · Answer

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.

Answer: D

salphonso · Answer

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?

EMPOWERgmatRichC · Answer

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

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

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