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
.
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.
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