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Appearance. In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest common divisor of x and y is denoted . For example, the GCD of 8 and 12 is 4, that is ...
hide. In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers. In the important case of univariate polynomials over a field the ...
Fermat's factorization method. Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: Each odd number has such a representation. Indeed, if is a factorization of N, then. Since N is odd, then c and d are also odd, so those halves are integers.
A key advantage of the Euclidean algorithm is that it can find the GCD efficiently without having to compute the prime factors. [9] [10] Factorization of large integers is believed to be a computationally very difficult problem, and the security of many widely used cryptographic protocols is based upon its infeasibility. [11]
In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout who proved it for polynomials, is the following theorem: Bézout's identity — Let a and b be integers with greatest common divisor d. Then there exist integers x and y such that ax + by = d. Moreover, the integers of the form az + bt are exactly the ...
Sunzi's original formulation: x ≡ 2 (mod 3) ≡ 3 (mod 5) ≡ 2 (mod 7) with the solution x = 23 + 105k, with k an integer In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition ...
Four principles. How to Solve It suggests the following steps when solving a mathematical problem: First, you have to understand the problem. [ 2 ] After understanding, make a plan. [ 3 ] Carry out the plan. [ 4 ] Look back on your work. [ 5 ]
Lagrange's four-square theorem, also known as Bachet's conjecture, states that every nonnegative integer can be represented as a sum of four non-negative integer squares. 1 That is, the squares form an additive basis of order four. where the four numbers are integers. For illustration, 3, 31, and 310 can be represented as the sum of four ...
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