tdkk123 wrote:The product of all prime numbers from 2 to 29 plus 1 equals K. Which of the following is true?
I. K is divisible by prime number.
II. K has a prime factor greater than 29.
III. P is divisible by 30

OA [spoiler]I & II[/spoiler]

I undertand how I but can someone explain how II is possible too?

Let's, for convenience, use the symbol "29$" to mean "the product of the primes up to 29". The question asks about the divisors of 29$ + 1.

First, 29$ + 1 is some large positive integer. Every positive integer greater than 1 is divisible by at least one prime, so I must be true.

Second, 29$ is divisible by *every* prime up to 29. For example, 29$ is a multiple of 13. When we add 1, the number 29$ + 1 is now 1 greater than a multiple of 13. That's just another way of saying that the remainder is 1 when we divide 29$ + 1 by 13. So, when you divide 29$ + 1 by any prime up to 29, the remainder will always be 1, and 29$ + 1 is thus not divisible by any prime number up to 29. Since 29$ + 1 must have at least one prime divisor, that prime divisor will need to be larger than 29, so II must be true.

For III, I don't know what P is supposed to represent here (it's not mentioned anywhere else). If P is equal to 29$ + 1, then III is certainly not true; as discussed above, 29$ + 1 is not divisible by any of the primes 2, 3 or 5, so it certainly is not divisible by 30.