chieftang wrote:Let P = 200! + 1.
Then, the greatest prime factor of P must be:
(A) Less than 40
(B) Between 40 and 80
(C) Between 80 and 120
(D) Between 120 and 160
(E) Greater than 160
Source: GMAT Hints
Since the difference between them is 1, P and 200! are consecutive integers. Consecutive integers are
COPRIMES: they share no factors other than 1.
Let's examine why:
If x is a multiple of 2, the next largest multiple of 2 is x+2.
If x is a multiple of 3, the next largest multiple of 3 is x+3.
Using this logic, if we go from x to x+1, we get only to the next largest multiple of 1. So 1 is the only factor common to both x and x+1. In other words, any two consecutive integers -- including P and 200! -- are COPRIMES.
200! = 1 * 2 * 3 * 4...198 * 199 * 200.
Every prime number less than 200 is a factor of 200!.
Since P and 200! are COPRIMES -- meaning they share no factors other than 1 -- no prime factor less than 200 can be a factor of P.
Thus, the smallest prime factor of P must be greater than 200.
Since the smallest prime factor of P is greater than 200, so must be the greatest prime factor of P.
The correct answer is
E.
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