The number of points, having both coordinates as integers, that lie in the interior of the triangle with vertices (0, 0), (0, 31), and (31, 0) is

The vertices of the triangle are (0,0), (31,0), and (0,31). Thus, this is a right-angled triangle, whose equation of hypotenuse is given by $$ x+y=31$$. Now, we need to find the points whose both coordinates are integers, and lie inside the triangle. 

This means that - $$x>0$$, $$y>0$$, and $$x+y<31$$.

$$x+y<31$$ => $$y<31-x$$ and $$y>0$$ => $$0<y<31-x$$

If x = 0, then y can take 30 values {1,2,3,...,30}
If x = 1, then y can take 29 values {1,2,3,...,29}
If x = 2, then y can take 28 values {1,2,3,...,28}
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If x = 29, then y can take only one value {1}

Thus, the total number of points inside the triangle = $$1+2+3+.....+30$$ = $$\dfrac{30\times 31}{2}=435$$ points.