The Remainder Theorem

The Remainder Theorem is a key proposition in elementary Number Theory. It is the basis for radix representation undergirding the familiar decimal notation which permits the universal school procedures for addition, subtraction, multiplication, and division of integers. It offers Euclidean algorithm for computing the greatest common divisor of two integers. The Theorem underlies modular arithmetic, required for solving linear congruence equation; enables proofs of Fermat‘s Theorem, Euler’s Theorem, Wilson’s Theorem, and the Fundamental Theorem of Arithmetic; and offers continued fractions for number representation and for solving linear Diophantine equations in two unknowns. It is used in an elementary proof of the divergence of the infinite series representing the sum of reciprocals of all primes.