ヤコービとルジャンドルの記号

最近暗号化が基づく数学を学びながらヤコービとルジャンドルの記号に関して読みました。計算式は分かりますが質問が残っています。
The Legendre symbol, named for Adrien-Marie Legendre(France, 1752-1833), is defined as:
(a/p) = a^((p-1)/2)(mod p),
where a is an integer, p an odd prime, and (a/p) is the syntax for Legendre symbol (not division).
Thus, (a/p) = one of:
1. 0, if p divides a
2. 1, if a is a quadratic residue mod p
3. -1, if a is not a quadratic residue mod p
A quadratic residue mod p is a number whose remainder when it is divided by p is a perfect square (4,9,25,etc). For example, (39/23) = 1, as 39 mod 23 = 16, which is a perfect square.
The Jacobi symbol (a/n) is a generalization of the Legendre which does not require that n be prime (although it must be odd). If n has the prime factorization:
n = p1^e1*p2^e2*...*pk^ek, then (a/n) is defined as
(a/n) = (a/p1)^e1*(a/p2)^e2*...*(a/pk)^ek
It seems that each of the quantities in () is a Legendre symbol, as each pk is an odd prime.
What I do not understand is:
1. What the Jacobi symbol quantity indicates. Does it have significance regarding whether "a" is a quadratic residue mod n?
2. Why these two quantities are relevant to cryptography. Do they facilitate prime factorization or the computation of key values?