The fundamental idea in the study of divisibility is the notion of congruences. Two integers a and b are said to be congruent modulo m if the difference a-b is a multiple of m. Congruences can be added and multiplied and this leads to a great simplification oof many computations. e.g. we can compute without much difficulty the last three digits of 97 raised to the 1997 power.
We study the basic arithmetic properties of congruences, solutions of linear congruences, of simultaneous congruences via the Chinese Remainder Theorem, and higher degree congruences.
Applications of this include properties of card shuffles, divisibility properties of binomial coefficients, various criteria for divisibility, and methods for construction of magic squares.