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In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than the absolute value of the divisor. A fundamental property is that the quotient and the remainder ...
Euclid's lemma — If a prime p divides the product ab of two integers a and b, then p must divide at least one of those integers a or b. For example, if p = 19 , a = 133 , b = 143 , then ab = 133 × 143 = 19019 , and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as well.
Euclid's algorithm is widely used in practice, especially for small numbers, due to its simplicity. [118] For comparison, the efficiency of alternatives to Euclid's algorithm may be determined. One inefficient approach to finding the GCD of two natural numbers a and b is to calculate all their common divisors; the GCD is then the largest common ...
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
Euclid's lemma: if a prime number divides a product of two numbers, then it divides at least one of those two numbers; Euclidean domain, a ring in which Euclidean division may be defined, which allows Euclid's lemma to be true and the Euclidean algorithm and the extended Euclidean algorithm to work
Many other theorems in elementary number theory, such as Euclid's lemma or the Chinese remainder theorem, result from Bézout's identity. A Bézout domain is an integral domain in which Bézout's identity holds. In particular, Bézout's identity holds in principal ideal domains. Every theorem that results from Bézout's identity is thus true in ...
The fundamental theorem of arithmetic can also be proved without using Euclid's lemma. [13] The proof that follows is inspired by Euclid's original version of the Euclidean algorithm . Assume that s {\displaystyle s} is the smallest positive integer which is the product of prime numbers in two different ways.
In mathematics and other fields, [a] a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem " or an "auxiliary theorem".