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