An interesting (and famous) probability

Number Theory Level 3

Let \({P}_{N}\) be the probability that two randomly selected integers on the interval \(\left[ 1,N \right] \) are coprime. Evaluate \[\lim_{N \to \infty}{P}_{N}\]


Two integers \(a\) and \(b\) are coprime if and only if \(gcd\left( a,b \right) =1\)

Round your answer to \(3\) decimal places, and express as a decimal instead of a percent (Not \(92.4\)%, instead enter \(0.924\)).


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