This is a high level problem that is actually quite simple if you have a good command over number properties.

The first premise is 2*4*6*....*100+1

Lets simplify this by pulling out a 2,

now we have 2(1*2*3*4*...*50)+1

So, not using the +1, the product of all those numbers is something humongous. Lets make up a number 454,413

If 454,413 is divisible by 1,2,3,4...,50 then 454,414 can not be divisible by any of them (besides 1, but it does not matter as 1 is not prime). So the smallest possible prime that it is divisible by has to be larger than 50. Hence the answer is E.

Lets look at it in a simpler way. 18 is divisible by 6,3,2,9,18. But if you add 1 making the number 19, it is not divisible by any of those. So its factors (19,1) are larger than 18 (not counting 1 for original problem because it is not a prime #)