  If o and * are two binary operations in a number system, then three elements
a,b,c in that number system are said to satisfy the distributive property of
the operation o over the operation * if, ao(b*c)= (aob)*(aoc) Now, suppose that
the number system is the rationals,and the operations o and * are among the
four usual operations of addition, multiplication, subtraction, and division.
If we allow for o and * to be the same operation, then there are precisely 16
combinations with the operation o being one of the four usual operations in Q;
and likewise for the operation *. Two of these 16 combinations are when o is
the multiplication operationand * being the addition operation; and when is o
is multiplication and * is subtraction. For these two combinations, the above
stated distributive property is universally satisfied; that is, for ane three
rational numbers a,b,and c. In this work, we examine the other fourteen
combinations, to find out when the distributive property is satisfied. Of these
14 combinations or cases, eleven are easy/straightforward, in that almost
always, one of the three rational numbers a, b, c; must be zero or 1. The
remaining three cases or combinations are much more complicated, and number
theory is involved.
