A quadratic residue is a number that can be expressed as the square of an integer modulo a given number. In simpler terms, if you take an integer, square it, and then divide by another integer (the modulus), the remainder is called a quadratic residue. For example, when considering the modulus 7, the squares of integers 0 through 6 yield the residues 0, 1, 2, and 4. Quadratic residues are important in number theory and have applications in cryptography, particularly in algorithms like the RSA algorithm. They help in understanding the properties of numbers and their relationships, especially in modular arithmetic, which is a key concept in various mathematical fields.