Linear Congruences - lucedu
In ordinary algebra, an equation of the form ax = b (where a and b are given real numbers) is called a linear equation, and its solution x = b=a is obtained by multiplying both sides of the equation by a 1 = 1=a The subject of this lecture is how to solve any linear congruence ax b (mod m) where a;b are given integers and m is a given positive
13 Congruences - NIU
Alternatesolution: The equation reduces to the congruence 35x≡ 45 (mod 75) This simplifies to 7x≡9 (mod 15), and multiplying both sides by −2 gives x≡−3 (mod 15) Thus 75y= 45 +3(110) = 375 and so x= −3, y= 5 is a solution
33 Linear Congruence Equations
After solving a congruence equation, be sure to check your answers In the next example we will solve the congruence equation 7x ⌘ 22(mod 39) A solution to this congruence equation is given on the bottom of page 69 and the top of page 70 of the text The derivation given in the text introduces a “hidden” factor of 6 Since (6,39) = 3,
3 QUADRATIC CONGRUENCES
course infinitely many solutions But when we count solutions to a congruence equation we treat congruent solutions as the same So this quadratic has just two solutions, as usual But if the modulus is not prime here can be more than two solutions by virtue of t the fact that there can be more than two square roots The squares mod 27 are
Congruence of Integers
2 Congruence Equation Let m be a positive integer and let a;b 2 Z The equation ax · b mod m (1) is called a linear congruence equation Solving the linear congruence equation (1) is meant to flnd all integers x 2 Zsuch that mj(ax¡b) Proposition 5 Let d = gcd(a;m) The linear congruence equation (1) has a solution if and only if djb Proof
Linear Congruences Theorem 1
0 can be obtained by solving the congruence a g x b g (mod m g) This is possible since a g; m g = 1 (See last theorem ) Example 3 42x 12 (mod 78) 2 Proof 1 If ax
Math 8: Prime Factorization and Congruence
sition follows from the definition of congruence modulo m and our previous theorems about when d = sa+tb has solutions: Proposition 13 6 Let m ∈ N and let a,b ∈ Z The congruence equation ax ≡ b mod m has a solution x ∈ Z if and only if hcf(a,m) b Proof: Let d =hcf(a,m) We first prove the (⇒) direction
44 Solving Congruences using Inverses
Translating the last equation back into a congruence, we nd the solution to the simultaneous congruences, x 206 pmod 210q: Fermat’s Little Theorem One of the most useful of important discoveries of the great French mathemati-cian Pierre de Fermat is that p divides ap 1 1 whenever p is prime and a is an integer not divisible by p
Quadratic Congruences
quadratic congruence € ax2+bx+c≡0(modp) for an odd prime p (with (a,p) = 1) is equivalent to solving the simpler congruence € y2≡Δ(modp), where Δ € =b2−4ac (the discriminant of the quadratic) and x and y are related by the linear congruence € y≡2ax+b(modp) (Since (2a,p) = 1, we can recover x from y ) The same line of argument
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