Bezout's Identity (Bezout's lemma)
Bezout's Identity: Bezout's Identity, also known as Bezout's Lemma, is a fundamental theorem in number theory that describes a linear relationship between the greatest common divisor (GCD) of two integers and the integers themselves.
This theorem has significant implications and applications in various fields, including cryptography, algebra, and engineering.
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What is Bezout's Identity?
Bezout's Identity (Bezout's lemma) states that for any two integers a and b, there exists x and y such that:
ax + by = gcd(a,b)
where gcd(a,b) is the greatest common divisor of a and b.
Bezout's Identity Example
Consider the integers a = 56 and b = 98. The GCD of 56 and 98 is 14. Bezout's Identity guarantees the existence of integers x and y such that:
56x + 98y = 14
One possible solution is x = -3 and y = 2:
56(−3) + 98(2) = -168 + 196 = 28
Mathematical Formulation of Bezout's Identity
Let a and b be any integer and g be its greatest common divisor of a and b. Then, there exists integers x and y such that ax + by = g ...(1).
The pair (x, y) satisfying the above equation is not unique. However, all possible solutions can be calculated.
We can find x' and y' which satisfies (1) using Euclidean algorithms . All possible solutions of (1) are given by,
where k is any integer.
It is easy to see why this holds. Just plug in the solutions to (1) to have an intuition.
Also, it is important to see that for general equation of the form,
u=gcd(a, b) is the smallest positive integer for which ax+by=u has a solution with integral values of x and y.
Statement: If gcd(a, c)=1 and gcd(b, c)=1, then gcd(ab, c)=1.
Bezout's Identity Proof
Above Statement can be easily proved using Bezout's Identity.
ax+cy=1 and bu+cv=1
Multiply the above two equations,
(ax+cy)(bu+cv)=1
The above implies,
ab(ux)+c(axv+buy+cyv)=1
1 is the only integer dividing L.H.S and R.H.S .
Hence, gcd(ab, c) = 1.
Applications of Bezout's Identity in Engineering
Bezout's Identity has numerous applications in engineering and computer science, particularly in areas involving integer solutions and modular arithmetic.
1. Cryptography
In cryptography, Bezout's Identity is used in algorithms such as the RSA algorithm. It helps in finding multiplicative inverses in modular arithmetic, which is essential for encryption and decryption processes.
2. Control Systems
In control systems engineering, Bezout's Identity is used in the design of controllers. It helps in solving Diophantine equations that arise in system stability and performance criteria.
3. Coding Theory
In coding theory, Bezout's Identity aids in error detection and correction. It is used to construct linear codes and solve equations that ensure reliable data transmission.
4. Signal Processing
In signal processing, Bezout's Identity helps in designing filters and solving equations related to discrete signals. It is used in the analysis and synthesis of signals.
5. Algorithm Design
In computer science, Bezout's Identity is used in algorithm design, particularly for algorithms involving integer arithmetic and modular computations. It provides efficient ways to solve problems related to linear Diophantine equations.
Bezout's Identity Solved Examples
Example 1: Find the Bézout's Identity for a = 30 and b = 42.
Step 1: Calculate the GCD using the Euclidean algorithm.
42 = 1 × 30 + 12
30 = 2 × 12 + 6
12 = 2 × 6 + 0
So, gcd(30, 42) = 6
Step 2: Work backwards to express 6 as a linear combination of 30 and 42.
6 = 30 - 2 × 12
6 = 30 - 2 × (42 - 30)
6 = 30 - 2 × 42 + 2 × 30
6 = 3 × 30 - 2 × 42
Therefore, x = 3 and y = -2 in the Bézout's Identity: 3(30) + (-2)(42) = 6
Example 2: Find the Bézout's Identity for a = 48 and b = 18.
Step 1: Calculate the GCD using the Euclidean algorithm.
48 = 2 × 18 + 12
18 = 1 × 12 + 6
12 = 2 × 6 + 0
So, gcd(48, 18) = 6
Step 2: Work backwards to express 6 as a linear combination of 48 and 18.
6 = 18 - 12
6 = 18 - (48 - 2 × 18)
6 = 3 × 18 - 48
Therefore, x = -1 and y = 3 in the Bézout's Identity: (-1)(48) + 3(18) = 6
Example 3: Find the Bézout's Identity for a = 17 and b = 23.
Step 1: Calculate the GCD using the Euclidean algorithm.
23 = 1 × 17 + 6
17 = 2 × 6 + 5
6 = 1 × 5 + 1
5 = 5 × 1 + 0
So, gcd(17, 23) = 1
Step 2: Work backwards to express 1 as a linear combination of 17 and 23.
1 = 6 - 5
1 = (23 - 17) - (17 - 2 × 6)
1 = (23 - 17) - (17 - 2 × (23 - 17))
1 = (23 - 17) - (17 - 2 × 23 + 2 × 17)
1 = 23 - 17 - 17 + 2 × 23 - 2 × 17
1 = 3 × 23 - 4 × 17
Therefore, x = -4 and y = 3 in the Bézout's Identity: (-4)(17) + 3(23) = 1
Example 4 : Find the Bézout's Identity for a = 24 and b = 36.
For a = 24 and b = 36:
GCD(24, 36) = 12
Bézout's Identity: 12 = 3(24) + (-2)(36)
Example 5 : Determine x and y in the Bézout's Identity for a = 51 and b = 21.
For a = 51 and b = 21:
GCD(51, 21) = 3
Bézout's Identity: 3 = 2(21) + (-1)(51)
Example 6 : Express the GCD of 40 and 28 as a linear combination of these numbers.
For 40 and 28:
GCD(40, 28) = 4
Bézout's Identity: 4 = (-1)(40) + 2(28)
Example 7 : Find the Bézout's Identity for a = 19 and b = 29.
For a = 19 and b = 29:
GCD(19, 29) = 1
Bézout's Identity: 1 = 11(19) + (-7)(29)
Example 8 : Determine the coefficients in the Bézout's Identity for a = 84 and b = 30.
For a = 84 and b = 30:
GCD(84, 30) = 6
Bézout's Identity: 6 = 1(84) + (-2)(30)
Example 9 : Express the GCD of 68 and 45 as a linear combination of these numbers.
For 68 and 45:
GCD(68, 45) = 1
Bézout's Identity: 1 = 7(68) + (-11)(45)
Example 10 : Find the Bézout's Identity for a = 101 and b = 37.
For a = 101 and b = 37:
GCD(101, 37) = 1
Bézout's Identity: 1 = 8(37) + (-3)(101)
Practice Problems on Bezout's Identity (Bezout's lemma)
1. Find the Bézout's Identity for a = 63 and b = 42.
2. Express the GCD of 105 and 77 as a linear combination of these numbers.
3. Determine x and y in the Bézout's Identity for a = 91 and b = 35.
4. Find the Bézout's Identity for a = 18 and b = 25.
5. Express the GCD of 144 and 89 as a linear combination of these numbers.
6. Determine the coefficients in the Bézout's Identity for a = 70 and b = 48.
7. Find the Bézout's Identity for a = 111 and b = 201.
8. Express the GCD of 95 and 36 as a linear combination of these numbers.
9. Determine x and y in the Bézout's Identity for a = 169 and b = 130.
10. Find the Bézout's Identity for a = 187 and b = 143.
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Conclusion
Bézout's Identity is a fundamental theorem in number theory that establishes a profound connection between the greatest common divisor (GCD) of two integers and their linear combinations. It states that for any two integers a and b, their GCD can always be expressed as a linear combination of a and b, i.e., there exist integers x and y such that ax + by = gcd(a,b). This identity not only provides a deeper understanding of the relationship between numbers and their factors, but also has practical applications in various areas of mathematics and computer science, including cryptography, solving linear Diophantine equations, and the implementation of efficient algorithms for computing GCDs and modular arithmetic operations.
