Does the integer k have a factor p such that 1<p<k?

1. k>4!
2. 13!+2(<=)k(<=)13!+13

I thought the answer was E. The questions seems to be asking whether k is prime or not, right? How does statement 2 tell us that????? The OG guide doesn't do a good job of expaining the answers. Please help!!!

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Definitely a tough one.

You need to know this concept: any time you add numbers, any factors shared by those numbers will also be factors of the sum. (This is also true when subtracting.)

So 2+4=6. 2 has a factor 2. 4 has a factor 2. The sum (6) has a factor 2.
14+21=35. 14 has a factor 7. 21 has a factor 7. The sum (35) has a factor 7.

13! + 2 = 13*12*11*10*9*8*7*6*5*4*3*2 + 2. The first number has a factor 2. 2 has a factor 2. The sum has a factor 2. If this sum has a factor 2, it is not a prime number.

13! + 3 = 13*12*11*10*9*8*7*6*5*4*3*2 + 3. The first number has a factor 3. 3 has a factor 3. The sum has a factor 3. If this sum has a factor 3, it is not a prime number.

And so on all the way up to:
13! + 13 = 13*12*11*10*9*8*7*6*5*4*3*2 + 13. The first number has a factor 13. 3 has a factor 13. The sum has a factor 13. If this sum has a factor 13, it is not a prime number.

So none of the integers between 13!+2 and 13!+13 (inclusive) are prime. You can answer the given question Yes definitively or the rephrased question "is k prime?" No definitively.

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