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We say that two lattice points are visible from one another if there is no lattice point on the open line segment joining them.
Now, if {{math|1=gcd(''a'', ''b'') = ''d'' > 1}}, then writing ''a'' = ''da''<sup>2</sup>, ''b'' = ''db''<sup>2</sup> one observes that the point (''a''<sup>2</sup>, ''b''<sup>2</sup>) is on the line segment which joins (0, 0) to (''a'', ''b'') and hence (''a'', ''b'') is not visible from the origin. Thus (''a'', ''b'') is visible from the origin implies that (''a'', ''b'') = 1. Conversely, it is also easy to see that gcd(''a'', ''b'') = 1 implies that there is no other [[integer lattice point]] in the segment joining (0, 0) to (''a'', ''b'').
Thus, (''a'', ''b'') is visible from (0, 0) if and only if gcd(''a'', ''b'') = 1.
Notice that <math>\frac{\varphi(n)}{n}</math> is the probability of a random point on the square <math>\{(r,s)\in \mathbb{N} : \max(|r|,|s|)=n\}</math> to be visible from the origin.
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