Solving Linear Congruence: 12x \equiv 1(mod5)

  • Thread starter Ed Aboud
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In summary, the conversation is about solving the congruence 12x ≡ 1 (mod 5) using different methods. The first method involves finding the greatest common divisor (gcd) of 12 and 5, and then using Euclid's Algorithm to find the value of x. The second method involves using the formula x ≡ ba^(φ(m)-1) (mod m) to solve the congruence. The conversation also includes a discussion about reducing numbers (such as 12 (mod 5)) and whether it is valid to do so.
  • #1
Ed Aboud
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Hi all.

Can someone please tell me what is going wrong here.

Solve

[tex] 12x \equiv 1(mod5) [/tex]


[tex]gcd(12,5) = 1 [/tex]

By Euclid's Algorithm =>

[tex] 1 = 5.5 - 2.12 [/tex]

So r is 5 in this case.
[tex] x = r ( \frac{b}{d} )[/tex]

Where b is 1 and d = gcd(12,5) = 1
[tex] x = 5 ( \frac{1}{1} ) [/tex]

[tex] x = 5 [/tex]

Ok fair enough but then I solve the congruence using

[tex] x \equiv b a^\phi^(^m^)^-^1 (mod m) [/tex]

[tex] x \equiv (1) 12^3 (mod5) [/tex]

[tex] x \equiv 3 (mod 5 ) [/tex]

I know this is the correct solution but what did I do wrong in the other one.

Thanks for the help!
 
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  • #2
don't know if this is valid, but isn't the first expression equivalent to [tex]
2x \equiv 1(mod5)
[/tex]
then 2x = "6" mod 5

[tex]x\equiv 3(mod5)[/tex]
yes i know that one is not supposed to do division, but modulus is prime, and there is a multiplicative inverse that i multiplied by (3)
 
  • #3
Ok I'm not very good at this, but why is the first one equivalent to [tex] 2x \equiv 1(mod5) [/tex].

Did you reduce the 12 (mod 5) ? Are you able to do that?
 
  • #4
I think so as [tex]12\equiv 2(mod5)[/tex]
 

1. What is a linear congruence?

A linear congruence is a type of equation in which the unknown variable appears in both the numerator and denominator, and the goal is to find the value of the variable that satisfies the equation.

2. How do you solve a linear congruence?

To solve a linear congruence, you must first isolate the variable on one side of the equation. Then, use inverse operations to cancel out any constants or coefficients. Finally, divide both sides of the equation by the remaining coefficient to find the solution.

3. What does the notation "12x ≡ 1(mod5)" mean?

The notation "12x ≡ 1(mod5)" means that the remainder when 12x is divided by 5 is equal to 1. In other words, 12x leaves a remainder of 1 when divided by 5.

4. How many solutions can a linear congruence have?

A linear congruence can have either one solution, no solutions, or infinitely many solutions. The number of solutions depends on the values of the coefficients in the equation.

5. What is the purpose of solving a linear congruence?

Solving a linear congruence can be helpful in a variety of mathematical applications, such as calculating interest rates, determining the number of solutions to a system of equations, and finding the period of a repeating decimal or fraction.

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