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It computes a-1 in two steps: Firstly, it computes the greatest common divisor (gcd) of a and M. This part of the whole procedure used is called Euclidean Algorithm. Secondly, extending the Euclidean Algorithm finds the desired inverse a-1 of a MOD M. In the following section you will learn how the Euclidean Algorithm finds the gcd of a and M.
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Two integers are called relative prime if their greatest common divisor equals 1. Examples are: 4 and 5 are relatively prime because gcd(4,5)=1. So are 2 and 3, 2 and 5, 3 and 10, 26 and 27, 45 and 16. Counter examples are: 45 and 18 are not relative prime since gcd(45,18)=9 and not 1. 343 and 14 are not relative prime since gcd(343,14)=7.
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a(b means a does not divide b The Division Algorithm If a, b ( Z, b ( 0, then there are unique values of q, r ( N such that a = pb + r, where 0 ( r < greatest common divisor gcd of a and b is written as (a, b) Scholar p183 q32,33 If a = bq + r then (a, b) = (b, r) The Euclidean Algorithm repeated application of the division algorithm
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[DOC File]CIS 3362 Homework #2 - UCF Computer Science
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5) Use the Euclidean Algorithm to determine the greatest common divisor of 3077 and 2295. Please show all of your steps. Solution: Using the Euclidean Algorithm: gcd(a, b)=gcd(b, a mod b) 3077 = 1x2295 + 782. 2295 = 2x782 + 731. 782 = 1x731 + 51. 731 = 14x51 + 17. 51= 3x17, so the desired gcd …
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However, if n is large, the Euclidean Algorithm will again be more efficient in finding the multiplicative inverse of an element. Hence, if we obtain gcd(a, n) = 1 from the Euclidean Algorithm, we can use the steps involved to find the multiplicative inverse of a in . Algorithm 2: “Extended” Euclidean Algorithm
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Lamé’s Theorem: Let a,b(N (a(b). Then the number of divisions used by the Euclidean algorithm to find gcd(a,b) ( 5•decimal digits in b. We can recursively define sets, too, not just functions. There is a basis step and a recursion step with the possibility of an exclusion step. Definition: The set (* of strings over an alphabet ( is defined by
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The Euclidean Algorithm is a time tested efficient method to find the GCD of two integers, and it can easily be programmed to compute the number of assembly phases for a …
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[DOC File]Section 1: Rings and Fields - Radford
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The Euclidean Algorithm. The Euclidean Algorithm makes repeated use of the division algorithm to find the greatest common divisor of two positive integers. If we are given two positive integers a and b where , then if , then , If , then we compute. The last nonzero remainder, , is the greatest common divisor of a and b, that is, .
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Euclidean algorithm gcd calculator