This problem is one of my favorites - it's a huge number, so you'll probably want to test the concept with smaller numbers and then just extrapolate it to the larger one in question (a pretty good strategy on these big-number number properties problems in general).
Look at it this way: Every second number (2, 4, 6, 8, 10...) is divisible by 2. But if you add 1 to any of them, it's no longer divisible by 2 (3, 5, 7, etc.). You're off that "every second" cycle by adding 1.
Same thing for 3: every third number (3, 6, 9, 12...) is divisible by 3, but if you add 1 you're off that every third cycle (4, 7, 10, 13...).
Essentially, if you take a number and add 1 to it, it's no longer divisible by any of its previous factors (other than 1), as it's off of each of those cycles.
If you take the number 2*4*6*8...*50 and break it down to its prime factors, you know that it's divisible by every prime number up to 25. If you add 1 to it, it won't be divisible by those factors anymore. That's why the answer is E - it won't be divisible by any prime factors that are less than 25.