# Solving Modular Arithmetic: x\equiv2 (mod km)

• theIBnerd
In summary, the conversation discusses the topic of finding the result of x(mod km) when given two equations x(mod k) and x(mod m). It is determined that for this to be possible, k and m must be coprime. The conversation also includes a proposed solution using the equations x(mod k) and x(mod m), which is found to be valid. The conversation concludes with the realization that more work needs to be done.
theIBnerd
i might be making it up, but i am confused.

can we say:

$$x\equiv$$2 (mod k)
$$x\equiv$$2 (mod m)
hence
$$x\equiv$$2 (mod km) by km i mean k multiplied by m.

if not, what is the result? or can it be found?

No.

k=4
m=8
x=10
x=2(mod 4) and x=2(mod 8) but x=10(mod 32)

In general, if you have something mod m and something mod k, and want to discuss what happens mod mk, then you need a condition on m and k being coprime, or something similar.

i think i found sth:

say (k,m) = 1

x=a (mod k)
x=a (mod m)

x=kt+a and x=my+a
kt=my
t=mb
y=kb

then x=kmb+a
x-a=kmb
x-a=0 (mod km)
x=a (mod km)

it is valid, isn't it? any counterexamples?

That looks pretty good to me

:) then my problem is solved. now i should get back to work.

Yeah, I hate when that happens!

## 1. What is modular arithmetic?

Modular arithmetic is a mathematical system that deals with integers and their remainders when divided by a fixed number, called the modulus. It is often used to solve problems involving periodic or cyclical phenomena.

## 2. How do you solve modular equations?

To solve a modular equation such as x≡2 (mod km), you first need to find the value of k and m. Then, you can use the modulus operation to find all possible values of x that satisfy the equation. In this case, x can be any integer that leaves a remainder of 2 when divided by the product of k and m.

## 3. What does the notation "≡" mean in modular arithmetic?

The symbol "≡" represents modular equivalence, which means that two numbers have the same remainder when divided by a given modulus. For example, 7≡1 (mod 3) because both 7 and 1 leave a remainder of 1 when divided by 3.

## 4. Can modular equations have multiple solutions?

Yes, modular equations can have multiple solutions. In fact, there are infinitely many solutions for most modular equations. For example, x≡2 (mod 6) has the solutions x=2, 8, 14, 20, and so on.

## 5. How is modular arithmetic used in real-life applications?

Modular arithmetic has many practical applications, such as in computer science, cryptography, and scheduling. It is also used in fields such as music theory, where it can be used to find patterns in musical rhythms and melodies.

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