The purpose of this problem is to demonstrate that the probability that two random numbers are relatively prime is about 0.6. a. Let \(P=\operatorname{Pr}[\operatorname{gcd}(a, b)=1]\). Show that \(P=\operatorname{Pr}[\operatorname{gcd}(a, b)=d]=P / d^{2}\). Hint: Consider the quantity \(\operatorname{gcd}\left(\frac{a}{d}, \frac{b}{d}ight)\).

b. The sum of the result of part (a) over all possible values of \(d\) is 1 . That is \(\Sigma^{d \geq 1} \operatorname{Pr}[\operatorname{gcd}(a, b)=d]=1\). Use this equality to determine the value of P. Hint: Use the identity \(\sum_{i=1}^{\infty} \frac{1}{i^{2}}=\frac{\pi^{2}}{6}\).