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Extended Euclidean Algorithm for Modular Inverse
12. You have to write 1 = 240x + 17y so 240x ≡ 1 (mod 17) The Euclidean algorithm applied to 240 and 17 gives 240 = 17 ⋅ 14 + 2 17 = 2 ⋅ 8 + 1 The successive remainders are colored red. Now start from the top: 2 = 240 − 17 ⋅ 14 Go one line down: 1 = 17 − 2 ⋅ 8 Substitute the value you have for 2: 1 = 17 − (240 − 17 ⋅ 14) ⋅ ...
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Extended Euclidean Algorithm, what is our answer?
Here is a layout for the Extended Euclidean algorithm, so you don't haveto go back through the successive divisions. It is based on the fact that not only the g.c.d. (last non-zero remainder in the successive divisions) satisfies a Bézout relation, but all successive remainders, hence these coefficients are computed recursively:
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Extended Euclidean Algorithm: backward and forward form
That's usually not an issue for me because I teach least magnitude residue systems before the extended Euclidean algorithm. The augmented-matrix method works for any remainder sequence (or any sequence of linear transformations - just as does the analogous augmentation method in linear algebra, e.g. sequences of row-reductions). $\endgroup$
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elementary number theory - Proof of Extended Euclidean Algorithm ...
This process is repeated until we have rn = ax + by where x and y are integers. This reasoning leads us to conclude that gcd (a, b) = ax + by. Example: Using the Euclideaan Algorithm let's compute the gcd (68, 20) = 4: Using the procedures discussed above, we have 4 = 20 − 8 ∗ 2 and 8 = 68 − 20 ∗ 3.
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Extended euclidian algorithm - Mathematics Stack Exchange
As for why, the extended euclidean algorithm decreases degree of result by at least 1, but in cases where the degree decreases by more than 1 in a single step, you'd have multiple sums = 0. $\endgroup$
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elementary number theory - How does the (extended) Euclidean algorithm ...
This is a special-case of Hermite/Smith row/column reduction of matrices to triangular/diagonal normal form, using the division/Euclidean algorithm to reduce entries modulo pivots. Though one can understand this knowing only the analogous linear algebra elimination techniques, it will become clearer when one studies modules - which, informally, generalize vector spaces by allowing coefficients ...
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Extended Euclidean Algorithm: why does it work?
So the greatest common divisor of 314 and 136 is 2. This iterative method of finding the greatest common divisor of two integers is called the Euclidean Algorithm. I'm fine with the first line. If I have an integer combination 314x + 136y, I can use the first given equation to write instead 314x + 136y = (2 ⋅ 136 + 42)x + 136y = 2x ⋅ 136 ...
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Method of solving extended Euclidean algorithm for three numbers ...
The method to solve for x x, y y, and z z is essentially the same as solving for x x and y y given a fixed value of z z. The equation given is: ax + by + cz = n a x + b y + c z = n. This reduces to: ax + by = n − cz a x + b y = n − c z. The only restriction on z z is that gcd(a, b) ∣ (n − cz) gcd (a, b) ∣ (n − c z).
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What's the difference between the euclidean algorithm and the extended ...
What does the euclidean algorithm compute, and what problems is the extended euclidean algorithm used for? Can someone please show how they each differ on the pair $(210,65)$ number-theory
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RSA and extended euclidian algorithm - Mathematics Stack Exchange
To encrypt, you choose two prime numbers p p and q q, and in exponent e e coprime to φ(pq) = (p − 1)(q − 1) φ (p q) = (p − 1) (q − 1). Let's denote n n the cipher modulus. The exponent e e is invertible modulo φ(n) φ (n); let f f be its inverse (it can be found by the extended euclidean algorithm).
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