Imagine that you are an employee of Prime Construction, a company that specializes in building structures based on prime numbers and other mathematical peculiarities. Today, you are tasked with building a brick wall. Each brick has the dimensions \(1\times1\times p\) where \(p\) is a prime number. You have plenty of each size of brick, but building a wall with just one size of brick would be boring. In fact, building a wall with any two rows the same would be boring, and Prime Construction won't stand for that! If your brick wall must be 100 units tall, and each brick is laid with its longest side parallel to the ground, what is the minimum length the wall can have?
Details and Assumptions
No two rows can have the same exact combination of bricks. For example, if you were building a wall of length \(5\), if you laid down a brick of length \(2\) and then one of length \(3\), you couldn't make another row by laying down a \(3\) then a \(2\).
The wall must be a perfect rectangular prism with the dimensions \(1\times100\times N\) where \(N\) is your answer, AKA the minimum length of the wall.
You can't cut the bricks