Why a Number Is Prime
Move the slider to see that if is a prime number, squares cannot be arranged into a rectangular array unless the width or length is 1. That is, it is not possible to represent a prime as the product of two integers with .[more]
Let and be the quotient and remainder of the division of by . (That is, for each and , let , where and are positive integers and .) This Demonstration shows as a rectangle of blue squares plus an additional red squares.
If is not prime, there may be some that make red squares appear, but that only means that particular does not divide ; there are other that do divide , in which case no red squares appear.[less]