Motivation of Jacobi symbol

I fancy the motivation of introducing Legendre symbol is to simplify the quadratic residue check.

But, for Jacobi symbol, even when its value is 1, it doesn’t necessary means that its Legendre Symbol is 1: $$left(frac 2 {15}right)_{Jacobi} = left(frac 2 3right)left(frac 2 5right) =-1*-1 = 1$$
, and 2 is not quadratic residue of 15.

So what is the motivation of introducing Jacobi symbol?

Wiki introduced its usage in Primality_testing , "This is the basis for the probabilistic Solovay–Strassen primality test and refinements such as the Baillie-PSW primality test and the Miller–Rabin primality test." Are there other usages?