problem solving;Factors

[ash4gmat]

Hi,can someone help me find out and explain the solution for the below mentioned problem.

Q)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?

51

54

72

162

50!+2

[DavidG@VeritasPrep]

Hi,can someone help me find out and explain the solution for the below mentioned problem.

Q)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?

51

54

72

162

50!+2

Tricky problem. We know that 50! will contain all the primes that are between 1 and 50, right? So the first prime that 50! does NOT contain will be 53. (51 and 52 are not prime. Moreover, 51 is a factor of 50! because 51 = 3*17 and both 3 and 17 are captured in 50! 52 is a factor of 50! because 52 = 4 * 13, and both 4 and 13 are captured in 50!) If the smallest prime not captured in 50! is 53, then the smallest non-prime that isn't a factor of 50! would be 53*2 = 106. Put another way, 106 is the smallest non-prime number that has 53 as a factor. The factors of 106 are: 1, 2, 53, 106. The sum: 1 + 2 + 53 + 106 = 162. Answer is D