Modular arithmetic with a variable modulus and fractions

In summary, the conversation discusses solving a congruence for x and the use of fractions in modular arithmetic. It is mentioned that fractions are only defined when the denominator is prime to the base or when the HCF of the denominator and base divides the numerator. The conversation also suggests multiplying out the equation to solve it.
  • #1
Floating Info
8
0
(This is my first post.)

I can't seem to find a good way of solving this sort of congruence for x:

x^2 / 3 + 11 [itex]\equiv[/itex] 5 (mod x)

Through trial and error it appears at least 3 and 6 are answers, but how can you reach them regularly? (I'm heard conflicting things about fractions being defined for modular arithmetic. It might be that this isn't even a createable congruence.)
 
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  • #2
Floating Info said:
I'm heard conflicting things about fractions being defined for modular arithmetic.
If the denominator is prime to the base then it's always defined. Otherwise, only when the HCF of denominator and base happens to divide the numerator in the ordinary way:
1/3 (6) does not exist because there is no number x s.t. 3x[itex]\equiv[/itex]1 (6).
6/15 (9) is ok because HCF(15,9) = 3, which cancels to produce 2/5 (9) = 4.
Floating Info said:
x^2 / 3 + 11 [itex]\equiv[/itex] 5 (mod x)
Just multiply it out and see what you get.
 

1. What is modular arithmetic with a variable modulus?

Modular arithmetic with a variable modulus is a mathematical concept where the modulus (or divisor) changes in each step of the calculation. This allows for a more flexible and versatile approach to solving equations involving remainders.

2. How is modular arithmetic with a variable modulus different from regular modular arithmetic?

In regular modular arithmetic, the modulus remains constant throughout the calculation. With a variable modulus, the modulus changes based on the given equation, providing a more dynamic solution.

3. Can modular arithmetic with a variable modulus be used with fractions?

Yes, modular arithmetic with a variable modulus can be used with fractions. In fact, fractions often arise in modular arithmetic when dealing with remainders.

4. What are the practical applications of modular arithmetic with a variable modulus?

Modular arithmetic with a variable modulus has various applications in fields such as computer science, cryptography, and engineering. It can also be used to solve real-world problems involving remainders, such as scheduling tasks or calculating interest.

5. How can I solve equations involving modular arithmetic with a variable modulus?

To solve equations involving modular arithmetic with a variable modulus, you can use various methods such as the Chinese Remainder Theorem or the Extended Euclidean Algorithm. It is important to understand the properties and rules of modular arithmetic to successfully solve these equations.

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