The number of points with integral coordinates that are interior to the circle $${x^2} + {y^2} = 16$$   is :

Solution :
The number of points is equal to the number of integral solutions $$\left( {x,\,y} \right)$$  such that $${x^2} + {y^2} < 16.$$   So $$x,\,y$$  are integers such that $$ - 3\, \leqslant x \leqslant 3,\, - 3 \leqslant y \leqslant 3$$      satisfying the inequation $${x^2} + {y^2} < 16.$$   The number of selections of values of $$x$$ is 7, namely $$ - 3,\, - 2,\, - 1,\,0,\,1,\,2,\,3.$$     The same is true for $$y.$$ So the number of ordered pairs $$\left( {x,\,y} \right)$$  is $$7 \times 7.$$  But $$\left( {3,\,3} \right),\,\left( {3,\, - 3} \right),\,\left( { - 3,\,3} \right),\,\left( { - 3,\, - 3} \right)$$       are rejected because they do not satisfy the inequation $${x^2} + {y^2} < 16.$$   So, the number of points is 45.