Set S contains points whose abscissa and ordinate are both natural numbers. Point P, an element in set S has the property that the sum of the distances from point P to the point (8, 0) and the point (0, 12) is the lowest among all elements in set S. How many such points P exist in set S?  Choose the correct choice from the following answers
A) 1
B) 5
C) 11
D) 8
E) 3
Explanatory Answer   The sum of the distances from point P to the other two points will be at its lowest only when point P lies on the line segment joining the points (8, 0) and (0, 12).   The equation of the line segment joining the points (8, 0) and (0, 12) is x/8 + y/12 = 1   Or the equation is 12x + 8y = 96 or 3x + 2y = 24.   The question states that the elements of set S contain points whose abscissa and ordinate are both natural numbers.   The equation of the line is 3x + 2y = 24 and hence, x will take even values while y will take values that are multiples of 3.   The values are x = 2, y = 9; x = 4, y = 6; x = 6, y = 3.   Hence, there are 3 such points that exist in set S.   The correct choice is (3)
<result>E</result>
Set S contains points whose abscissa and ordinate are both natural numbers. Point P, an element in set S has the property that the sum of the distances from point P to the point (8, 0) and the point (0, 12) is the lowest among all elements in set S. How many such points P exist in set S?