Some Related Functions to Integer GCD and Coprimality

We generalize a formula of B. Litow [7] and propose several new formula linked with the parallel Integer Coprimality, Integer GCD and Modular Inverse problems as well. Particularly, we find a new trigonometrical definition of the GCD of two integers a, b ≥ 1: gcd(a, b) = 1 π ∫ π cos [ (b − a)x] 0 sin 2 (abx) sin(ax) sin(bx) dx. We also suggest a generalization of the GCD function to real numbers